Place garlic into a food processor with the motor running and process until chopped.
Add the basil, pine nuts, and salt.
Pulse until the basil and pine nuts are coarsely chopped; continue processing until finely chopped.
Drizzle in the olive oil with the motor running.
Transfer to a small bowl and stir in the grated cheeses.
Preheat the oven to 400° F.
Bring about 6 quarts of water to a boil and add about 2 tablespoons salt (optional).
Prepare a bowl of ice water.
Trim the asparagus and boil for 1 minute. Use tongs to remove the asparagus from the hot water. Reserve asparagus cooking water.
Refresh asparagus in the ice bath for 30 seconds. Use tongs to transfer the asparagus to a colander, drain well, and cut each spear in half crosswise and set aside.
Heat olive oil over medium heat in a 12 to 14 inch sauté pan. Add the eggplant and cook until golden brown and soft, about 5 minutes. Drain on a paper towel and set aside.
Cook the lasagne sheets in the same boiling water as the asparagus and cook for 2 minutes less than the package instructions. Drain and separate.
Coat a 13 x 9 inch baking dish with cooking spray.
Layer 4 sheets of lasagne on the bottom of the dish; top with 6 pieces of asparagus, 3 to 4 slices of eggplant and 3 tablespoons of Béchamel sauce.
Top with 4 sheets of lasagne, asparagus and eggplant as above, and 3 tablespoons of pesto sauce.
Repeat these 2 layers for a total of 4 layers. End with 4 sheets of lasagne, remaining Béchamel sauce and grated cheese. Sprinkle with bread crumbs.
Place the baking pan on a baking sheet and bake for 20 to 25 minutes, or until bubbling and golden brown on top. Let rest for 10 minutes before serving.