1. Preheat the oven to 375° F. Grease a 9×13 inch dish.
2. In a medium saucepan, melt together the olive oil and butter. Add the shallots, garlic, dried thyme, dried rosemary, kosher salt, and black pepper. Cook 5 minutes or until fragrant. Remove the garlic cloves and discard.
3. Whisk in the flour and cook for about 1 minute. Add the milk and broth. Bring to a boil and stir for 2-3 minutes. Remove from the heat and stir in the provolone and 1/2 cup of parmesan cheese. Stir until the cheese is fully melted and the sauce is smooth.
4. To assemble, spread 1/4 of the cheese sauce into the bottom of the prepared baking dish. Top with 3-4 lasagna sheets. Spread with 1/2 the pesto and sprinkle over some spinach. Add another 1/4 of the cheese sauce.
5. Place another 3-4 lasagna noodles on top, add the remaining pesto, spinach, and then another 1/4 of the cheese sauce, sprinkle over the remaining spinach.
6. Layer the remaining lasagna noodles and pour the remaining cheese sauce over top. Top with a 1/2 cup of parmesan cheese.
7. Bake, covered, for 30 minutes or until the cheese is bubbling. Uncover and bake another 15 minutes, until the cheese is browned. Let stand 10 minutes before serving.
