In food processor, pulse together dandelion greens, basil, walnuts, garlic, lemon zest, and lemon juice until well combined and everything is pulverized. Add Parmesan and then, with machine running, drizzle oil through top feed tube. If needed, stop the machine to scrape down sides of bowl.
Heat oven broiler. In large bowl, beat together eggs and milk. Gently stir in 1/4 cup pesto, trout, and sun-dried tomatoes.
In ovenproof 10-inch skillet over medium, heat oil. Add shallots; heat until softened, about 2 minutes. Pour in egg mixture so it’s evenly distributed in pan and heat until edges begin to set, occasionally lifting cooked eggs around edges with spatula, about 5 minutes. Distribute goat cheese overtop and place skillet in oven. Broil until center of frittata is set and cheese has browned, about 3 minutes.