It looks to me like the shelf is laminated together from several thicknesses of plywood (which makes for a strong shelf), and that the rectangular area just above the double-headed arrow has been rabbeted out to accommodate a bracket that is bolted to the wall and then hidden when that shelf is slid over it. I'm sure there are at least two of these rabbets per shelf. The bracket is probably a small steel plate with a support arm welded to the center and holes for bolts on either side. That rabbet is a couple of inches wide because that's how wide the plate needs to be to accommodate the arm and two holes. That means that, during construction, the contractor would have doubled or tripled up the studs in each of the bracket areas to accommodate bracket plates that are a couple of inches wide. When dealing with single studs, the edge is 1-1/2" or less, so you can't securely get two lag bolts into it side-by-side; a plate with only one bolt would be iffy. It's a nice installation, but one that needs to be planned for before the sheetrock has gone up.

opaone: I trust you mean in the sense of capturing effluent headed upward and outward such that it would miss a 27-inch entry aperture boundary by 3 inches. Weifei Zheng: A four-inch high shelf hood really won't have a significant buffer volume (assuming you mean volume below the baffles); the baffle assembly itself might be 1.5 inches thick and will have to be low as possible. The air flow in the volume above the baffles will not have any lingering fumes. But to get the flow at the baffles to be roughly equal across the baffle area, the pressure inside this type of hood has to be low relative to below the baffles, and this pressure loss moves the user up and to the left on the fan curve, reducing volumetric flow rate. Hence the suggestion for zero static pressure rated flow rate to be twice the required operating flow rate. (Also note that I did the calculation for a nominal 24-inch entry aperture (front to back); a 27-inch will require 1/8 more air flow to keep the velocity at 90 ft/min. Maximum stud space ducts of 3.5 x 14 might be advised, along with nearly matching slots in the hood rear.)

We figured however we put it into the cabinets, those holes are easier to fill than tile holes. I don't mean to be adversarial, but I disagree completely. Replacing a tile is a 15 minute and $15 DIY job, getting a paint and sheen match on cabinets is something I would not even try and I have years of experience doing it. I would just respray the entire sides of your cabinet as that would take 1/10th the time of trying to match them. However, the problem isn't drilling through your tile and that wouldn't work either. You need to accomplish what you want without drilling through the sides of your cabinets. Which can easily be done. You can make it look like it was a planned part of the kitchen and not an afterthought while preserving the sides of your cabinets. Or something like this..

I tend to agree about the hood. My partner has a tendency to make impulse decisions in the name of “funky” which led us to this hood. it was also intended to help tie in the black range which might have felt out of place with a stainless hood. We could probably replace it but then I feel like the black range will feel out of place. The soffit we tried to remove but it vents heat and we have to keep it. I do like the idea of potentially no island light. The original plan was some large funky pendants to anchor the space but that clearly deviated as well. I might try to go back to that. Same with the stove- I wanted stainless but we got this black one for an absolute steal. The cabinets on the pantry wall are a laminate wood color. We could try to match those on the stove wall with some shelves....I wanted to just put greenery on them- some cute grass, herbs, things to freshen up the place. Down the line, maybe we could replace the island as well. All white is way more my style. I’m starting to realize what everyone says: decisions to save money upfront might cost money in the long run. Maybe we can remove the pendant, paint the window wall, get some shelves, and bring in the stools, rugs, wicker baskets, etc and see how it’s feeling. Worst case, it’s...functional...and a lesson learned...