Prepare the pesto (this can be done the day before). Combine the pesto ingredients in a blender. Pulse until you reach the desired consistency.
Sauté asparagus pieces for 3 minutes on high heat, until you see it begin to brown. Take off heat and transfer to a separate bowl, stirring in 1-2 Tbsp. of the pesto (depending on how strong you want it).
Heat skillet or panini maker to medium (we really like cast iron for this if it’s available, but if not, use a non-stick pan).
Assemble the sandwich: one slice of Havarti, top with the prosciutto, then the pesto coated asparagus, and more slices of cheese. Top with the second slice of bread after spreading a thin layer of mayo on it. If you’re making multiple grilled cheeses, assemble them all before you start griddling.
Place the sandwich mayo side down in the skillet. Spread mayo on the bread slice now facing up and cook 3 minutes, or until bread is toasty brown. Turn and repeat. Your stove top and pan may require slightly different cooking times.