I don't have any experience with the Bluestar insert, but we did have a Zephyr. Below is the text of my post from another thread describing our experience. In a nutshell, we got rid of it and put in a remote blower. If there's ANY way for you to get the fan out of the kitchen, do it. I can't comment on those specific models, but we had a Zephyr Monsoon II model AK9346AS installed about a year ago and it we found it to be impossibly loud. Their literature says the sone level on low is 3.5, which, based on this chart, is about 45 decibels. I downloaded a decibel meter to my phone and with the phone sitting on the stove 35 inches below, the noise registered 80-82 db. That's MUCH louder than advertised, and we found it very uncomfortable and annoying--couldn't hear the Sonos speaker 12 feet away or anyone talking in a normal voice, and there was a sigh of relief whenever we turned the fan off. We eventually decided to switch it out and install a Wolf liner (with variable speed knob), an inline silencer, and 1400 cfm Abbaka roof mounted blower. It's quieter on high than the Zephyr was on low. The three pieces are a more expensive proposition (even if we hadn't made the Zephyr mistake), but the result is so vastly more pleasant that I still smile when I turn on the fan. In a nutshell, if there's any way you can listen to the Zephyr and Best, preferably in a similar setting, do it. Don't trust the literature.

Whichever one has baffle filters (not mesh filters) and a canopy shape, not a flat shape. Quiet has to do with your ductwork from the hood to the outside. If it is narrow, like 6" diameter, it will be noisy, and it won't matter what brand of hood you buy with respect to the noise. Also, be realistic, that an exhaust hood DOES have a motor operating almost right in your face. Impossible to be really quiet. People have imaginary expectations of a hood being quiet.

There is always make-up air, the question is whether it is adequate. No air leaves the kitchen via the hood that didn't enter the kitchen. If the entry path is restrictive, then (a) there may be a safety hazard with respect to combustion appliances in contact with the kitchen air -- back-drafting carbon monoxide, and (b) the hood flow will be less than one wants. Whether 400 blower-rated CFM (in practice significantly less actual volumetric flow rate) is sufficient depends on cooking surface temperature and various details about the hood configuration; and one's tolerance for grease and odor. Ignoring edge effects between a liner and a cabinet, and using the Wolf 22.5 inch by 40.375 inch liner as an example, the effective entry area is assumed to be 6.3 sq. ft. I would recommend 90 ft/min air velocity if wok cooking or hot searing is intended. This requires 6.3 x 90 => 500 - 600 actual CFM. Depending on pressure loss in the hood baffles, the selected blower, and the pressure loss of the MUA system, a 750 - 900 CFM rated (that is, hanging in free air) blower is needed. You will need deliberate, but possibly only passive, MUA for this configuration. The most optimistic assumption about lower plume velocities from induction cooking would reduce these values to 2/3. That is, a blower rated at 500 - 600 CFM could work. Municipalities enforcing the IBC requirements on MUA will still require some MUA scheme, such as a duct with a powered damper. A smaller hood is not a good fix for cutting required CFM, because adequate plume capture could be unsuccessful in some cases. It may be worth adding, as @opaone has pointed out w.r.t. commercial hoods, if the capture or containment volume -- the reservoir -- below the baffles is larger than typical of residential hoods, then plume spillage can be resisted with a somewhat lower volumetric flow rate. I am not aware of any quantification of this effect, due I think to spreading of the plume over the entire baffle space, but in the case of an insert one has this option: namely, to put the insert farther up in the cabinet and extend sheet metal down to the cabinet bottom. I wouldn't want to bet that a 400 CFM rated blower would be adequate in this case, but it would be a possible option to avoid dealing with MUA from a code enforcement POV. Whether combustion appliances are thereby safe is dependent on too many factors to make a pronouncement.

All of you seem to be new to this topic, so please rummage around here reviewing 'hood' and 'MUA' threads. There is also the Hood FAQ with hopefully useful information. https://www.houzz.com/discussions/5161173/hood-faq#n=77 A few key facts: CFM is cubic feet per minute, and measures a volumetric air flow rate. For a given hood, noise will be proportional to some power of the CFM. Induction cooking plumes tend to be lower velocity than gas cooking plumes, so for most conventional cooking the velocity of the air pulled into the hood by the hood system blower can be lower. Without wok cooking or meat searing, this might be as low as 60 ft/min for the fastest rising plumes, whereas for the more vigorous cooking, particularly on powerful gas range tops, 90 ft/min should be aimed for. Hood intake velocity equals the actual blower CFM divided by the hood entry aperture in square feet. Hood entry aperture should overlap rising and expanding plumes. Actual CFM is well less than rated CFM. See all the comments on this in myriad threads. Noise is mainly due to blade tip turbulence and baffle aerodynamic turbulence, with a contribution from duct turbulence. In minimizing this, more distant blowers will be a bit quieter than nearby blowers, distant blowers beyond in-duct silencers will be quieter than without. Oversized (necessarily remote) blowers run at low speed will be much quieter than small blowers operating at top speed for the same CFM. Microwave oven pretend hood functions are generally less than inadequate. Feel free to ask if i missed a topic raised above.