Rent scaffolding an access the soffit area above the cabinets. Measure the bulkhead-allowable space to within one-eighth inch. Define the length, depth and height of each section, including a 1-inch or 3-inch lip for each soffit box, if you want overhang.
Draw the kitchen cabinets and new soffits on graph paper. Have one square represent 4 inches of actual space. Sketch the framework only allowing for drywall thickness to come out flush with cabinet fronts. Use five-eighths inch thick drywall for this space. Determine the thickness of any bead board or wood veneer, so that the framework dimensions accommodate this covering as well.
Select high-quality 2-by-4-inch boards, so the framework is square and true. Cut framing sections with a circular saw for the top and bottom plates, and short studs that fit along the soffit framing’s front. Place short studs on 16-inch centers. Connect all framework sections with 4-inch screws, so that the unit stays intact when lifted into place.
Locate ceiling rafters and wall studs with a stud finder. Secure the framework box for each soffit to the wall studs and rafters using 4-inch screws driven by an electric screwdriver. Make adjustments with wood shims and check all work with a carpenter's level. Have assistants lift the framework into place, so you will not rest any framework weight on cabinets. Use ladders or scaffolding to maneuver the framework into place.
Run any electrical wiring needed for the soffit or nearby wall lighting through the framework. Use an electrical conduit to route wiring for safety reasons. Route wiring to lighting above the sink or under cabinets through this bulkhead space, as one option, especially if changing wiring inside actual wall framing will be difficult.
Nail drywall or other covering to the soffit framework. Add metal capping to the drywall corners to get a sharp edge and protect the drywall. Finish the soffit spaces smoothly and add wood trim as needed.