While you could run a duct through the concrete or concrete block wall to the outside and thence upward to a cap using stainless steel duct in the exterior zone, or out through the exterior joist that rests on the sill plate, there are recirculating hoods. The only one that has been reported on extensively here and deemed adequate is the Vent-a-Hood ARS.

https://www.houzz.com/discussions/2347276/vent-a-hood-ductless-ars-range-hood-update#n=65

Broan may make one also. Significant filtration of grease and odor is needed. When combining recirculating with gas burners, there could be some slight hazard due to the combustion effluent not being fully cleaned, as well as possible carbon dioxide buildup. This might depend on the rate that basement air is replaced by other means.

A do-it-yourself system comprising a standard baffled hood followed by ducting to a stacked collection of filters might also work, but would require some research.

Given your intended cooking, I would suggest re-evaluating ducting to the outside.