Step 1Cook the broccoli and asparagus together in boiling water until tender, 5 to 7 minutes. Meanwhile, cook the pattypan squash in a separate saucepan of boiling water until tender, 5 to 7 minutes. Remove the squash from the pan using a slotted spoon and set it aside; add the bell pepper to the water and cook it until tender, 5 minutes. As soon as each vegetable is done, place it in a bowl of ice water to shock it, then drain it and set aside.
Step 2Beat the eggs in a large bowl for 1 minute. Add the cheese, pesto and cooked spaghetti and stir until thoroughly mixed.
Step 3Melt 1 tablespoon of the butter in a skillet over medium-low heat. Add the egg mixture. Let it cook without stirring. After 8 minutes, start checking the bottom of the frittata with a spatula; when it is lightly browned and the top is starting to set, place a large plate over the skillet and carefully flip the frittata onto the plate. (The eggs will run a little, but that's OK because you'll be turning the frittata back into the skillet.) Melt the remaining 1 tablespoon of butter in the skillet and quickly slide in the frittata from the plate. Cook until the frittata is firm and lightly browned, 8 to 10 more minutes. Gently cut into it on the side to make sure the center is done.
Step 4While the frittata cooks, heat 1 tablespoon of oil in a skillet over medium heat. Rub each mushroom with 1 teaspoon of oil and sprinkle it with salt and pepper. Fry the mushrooms until soft, turning halfway through, 10 to 12 minutes.
Step 5Remove the mushrooms, add 1 tablespoon of oil to the skillet and cook the garlic until aromatic, 2 to 3 minutes. Add the broccoli, asparagus, squash, bell pepper and salt and pepper to taste and cook until heated through, 4 to 5 minutes.
Step 6Turn the frittata onto a serving plate and cut into pieces. Slice the mushrooms and serve with frittata wedges and the vegetables.