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Cutting a cabinet in half is not an easy task. In most of these installs the cabinet above the microwave is bumped up to give a tiered look in the kitchen. This looks better if other cabinets do it as well. I have built custom cabinets when the space was not available for this bump up.
Another option is to "blank" the space with a sheet of cabinet ply. with or with out doors. many of these micro also have integrated vents which eat into any possible cabinet space above.. so the missing space is taken up with ducts.
We need more information on what your cabinets are. Solid panel doors, raised panel doors, foil type? All have their ways to be made smaller. Some easy, some cannot be done.
The problem with end grain glue joints is they suck up the glue more so the joints are often "starved" some carpenter will "size" the joint first by an application of glue to the ends...let that dry...then glue again and join.