Remove the greens from the carrots and reserve for later use.
Place carrots on a heavy baking sheet and drizzle with olive oil and a sprinkle of salt and pepper.
Cook for 20 minutes until they just begin to brown and blister.
For the carrot top pesto, place washed greens in the basin of a food processor with the blade attachment. Combine olive oil, garlic, and the juice of one lemon. Blitz until smooth, adding a little olive oil if it feels too “pulp-y.”
Shave the cheese super thin, set aside.
Warm a shallow, heavy pan over medium heat while you prepare the sandwiches. Butter one side of each slice of bread.
Lay flat and layer with cheese, then pesto, then 4-5 grilled carrots. It’s okay if the stems stick out. Finish with another layer of cheese, if desired, and the other slice of bread.
Place in the pan and grill on each side for 2-4 minutes until browned as you prefer.