No wood can be closer than 30" to the cooking surface. It's 36" with some ranges. So you need to read the specs of your range. And understand them. Most have a prohibition against using deeper than 13" cabinets directly against the hood. So you make the hood wider. You also need to know the width of the corbels so that they can be outside that combustible zone. A 36" cooking surface needs a minimum 42"wide vent above it. With your decorative corbels, I'd increase that to 48". You also need at least 27" of front to back depth in order to capture the cooking byproducts. Yea, you need a liner, and a blower. Start there and build the hood around the option that you choose. you need to understand about code, cabinet construction, and overall proportion and aesthetics to do this successfully. Or just choose from your cabinet company's predesigned hood and blower combinations that you know will work.

I have the same issue and am planning a different approach than what Fori described because I have concerns about the hood sticking out too far. I plan to build a bulkhead to mount the hood under instead of using the cabinet, and then we can mount the cabinet doors on the front of tje bulkhead with perhaps some filler as needed around the edges where the cabinet face frame would be. The cabinet is really useless anyway with the vent running through it. And I think the difference in cabinet depths will actually look nicer than having everything on one plane. In many designs, the hood cabinet actually projects out father than the cabinets on either side, so this is just the reverse.

Mechanical Q: What makeup air requirements are associated with vented kitchen range hoods? (IRC M1503.4, M1501.2, G2447.6)  Three code provisions can trigger the needful makeup air in this situation.The bottom line is that makeup air is always required for high-volume hoods (> 400 cfm) and may be required for lower volume hoods when natural-draft appliances are used.  M1503.4 requires makeup air for kitchen hoods which exhaust more than 400 cfm. The makeup airflow rate must equal the exhaust flow rate and an intake damper on the makeup air duct must be interlocked with the control for the fan.  M1501.2requires that“ducted exhaust systems”(such as kitchen hoods,bath fans, clothes dryers) not induce negative pressure in excess of negative 3 Pascals at the location of a natural- draft combustion appliance. This provision applies regardless of the rated airflow of the exhaust system.  G2447.6 reinforces M1501.2, more explicitly for vented kitchen hoods. This provision is independent of the rated airflow through the hood.  A pressure test is required to demonstrate that the depressurization level is within the 3Pascal limit. If the depressurization limit is exceeded, makeup air is one potential solution.  There is no requirement to temper makeup air.

Capture area (hood intake aperture area) should overlap the expanding cooking plumes. Assume at least 10 degrees from vertical for all points across the top area of all likely pots and pans to be used on any or the hobs, and add some for burner flames if applicable. This implies that the capture area has to grow with height, and doesn't account for plume deviations due to cross drafts and human motion turbulence. Capture areas smaller than this will likely miss capturing effluent and the effluent will spread into the house where it either condenses on walls and fabrics, or is eventually removed from air replacement. Numerous threads here discuss sizing the hood, sizing the flow rate to assure that even if the plumes are captured they don't reflect out of the hood, sizing the duct for best balance between grease collection due to impingement and grease collection due to condensation, blower rated CFM to achieve desired CFM, and make-up air (MUA). Quieting of the hood system can be achieved using a remote blower and intermediate silencer such as Fantech makes. A 6-inch duct should be limited to 400 CFM, so a 600 CFM rated blower might be appropriate given typical pressure losses and blower fan curves. 10 x 4 duct, from a pressure loss point of view is equivalent to 5.7-inch round duct. (IRC Table M1506.2). For a given blower restricted by duct size, it is probably better to use a larger hood and miss out on some extreme cooking capture due to insufficient CFM, then to miss out full plume capture on all cooking due to insufficient area to overlap the plumes.