Preheat the oven to 425 degrees F. Line a large baking sheet with foil and set aside.
Place the broccoli, mushrooms, and potatoes on the baking sheet and drizzle with 2 tablespoons of the olive oil. Season with 1 teaspoon each Kosher salt and ground black pepper, then toss to coat. Roast for 30 minutes, or until the potatoes are fork-tender, and the veggies are slightly crisp around the edges.
Meanwhile, scramble the eggs. Heat the remaining 1 tablespoon olive oil in a large skillet over medium heat. Add the eggs and milk to a large bowl, seasoning with 1 teaspoon each Kosher salt and ground black pepper. Whisk until pale & frothy. Add the eggs to the skillet. Soft-scramble the eggs. Use a wooden spoon or spatula to push them around the skillet as the cook, until the eggs are just set, about 2-3 minutes. The eggs should be very fluffy and slightly wet! Remove from the heat and set aside.
To assemble, pop the tortillas into the microwave for 30 seconds to warm them up. Pile each with eggs, roasted veggies, a sprinkling of shredded mozzarella. Squeeze 1 tablespoon of pesto over top. Roll them up, tucking the sides toward the center then roll the filling into the center of the burrito.
To serve immediately, place the assembled burritos on a second baking sheet. Pop them back in the oven to melt the cheese, about 5 minutes.
