Preheat oven to 400 F. Line a rimmed baking sheet with aluminum foil.
Chop broccoli heads into small, 1 inch pieces. Dice broccoli stems into smaller ¼ inch pieces.
Place broccoli and garlic onto the lined baking sheet and drizzle 1 Tbsp olive oil onto everything. Bake in the oven for 15 minutes. Remove the pan from oven and add your choice of nuts. Place back into the oven for another 5-8 minutes, or until broccoli is brown and frizzly on the edges.
Transfer everything from the sheet pan to a blender or food processor. Add the remaining pesto ingredients plus 1 Tbsp water. Blend until completely smooth. If the pesto is having trouble breaking down, add a tsp of water at a time and blend until the pesto is smooth. Taste for salt and pepper. Set aside for pasta or store in airtight container for future use!
Cook pasta in a large pot according to box directions. In the meantime, make crispy panko breadcrumbs: Add panko breadcrumbs, a pinch of salt, black pepper, and 1 Tbsp butter to a small saucepan on medium heat. Stir until panko is golden brown and transfer into another bowl immediately (to prevent burnt breadcrumbs).
Reserve ½ cup pasta water and drain pasta. Add the hot pasta, 1 remaining Tbsp butter, ⅓ cup pesto, and ¼ cup pasta water back to the pot. Toss everything together until the pesto, water, and butter have emulsified. Continue to add an extra Tbsp of pasta water at a time until the sauce is smooth, glossy, and clinging to the pasta.
Top with toasted breadcrumbs, freshly grated parmesan, and enjoy!