Preheat oven to 400°F. Line a baking sheet (or two) with foil.
Dice pepper, squash, and onion; peel and dice potato, cut broccoli into bite-sized pieces, and slice mushrooms in half. Try making all veggies similar in size.
Add all veggies to a large mixing bowl, along with olive oil, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Toss until coated evenly, then spread out onto a baking sheet. If the veggies begin to overlap, a second baking sheet may used. Bake for 30 minutes until soft and slightly browned, stirring half way through.
While veggies are roasting, cook pasta to al dente according to the package instructions. Drain and return to pan. Stir in butter.
In a small bowl, whisk together the milk, egg, paprika, and remaining salt and pepper. Add to the pasta, stir, and cook over medium until heated through--approximately 5 minutes.
Remove from heat and stir in cheeses until fully coated. Fold in the veggies.
Serve with sliced strawberries on the side.