You might talk a tin knocker into making a custom hood.

The inspiration pic is working becuase the beam is not directly above the cooktop I think in your case this will need to be custom hood made exactly for your situation and could be expensive. I have a question why did you not take the cabinets to the ceiling in such a small space?You are fortunate in that you can go up but that beam will create some issues. and the counter is not deep enough to add a down draft.

Guessing that the ceiling (attic floor) joists are transverse to the beam and are supported on it, one would have to tear into the ceiling and attic timber layout and build a box beam setup in the attic such that the transverse joists are hung from the appropriate member of the beam pair with Simpson hangers. Then there would be room within the box for your, say, 8-inch duct and no lonely beam in the kitchen. I think the box structure would have to extend from the beam support posts in the two walls depending on whether the joists are full width or separate on each side of the existing beam. I would use Microlam for the beams in the attic to improve stiffness, and a few cross ties between the two new beams would make the structure extremely resistant to bending moments.

In other words, the parts and materials you need exist, but installing them will be messy. Partial ceiling removal will be needed just to get the new beams into the attic. The wall posts may need to be uncovered at the top for any needed modifications. If the attic is served by a scuttle and not stairs, then a ceiling hole is the only practical way for a carpenter to operate.

You likely will be hanging the hood extension from the new beams, or wood tied to the new beams, so all the necessary requirements and architectural details need to be worked out before ravaging the area. A complete initial plan beats an on-going evolving plan.

How about something like this? Install an island hood with a cutout in the chimney for the beam. The part of the chimney that extends to the ceiling and contains the duct would be on the kitchen side. The duct inside the chimney would be rectangular. The side of the chimney opposite the kitchen would be flush with the side of the beam.

-- amateur

Hi Wendy

I am wondering what you ended up doing here as we are dealing with the same issue now. Thank you!