Make sauce. Slowly whisk milk into cornstarch in small bowl, trying to avoid lumps.
Heat 1 T butter and tomato paste in medium pot over medium heat. Add stock, root beer, red pepper, and salt & pepper. Once incorporated, pour in milk & cornstarch mixture. Let cook until slightly thickened, about 5 minutes.
Preheat oven to 425F.
Lay asparagus on large baking sheet. Drizzle with olive oil and salt & pepper. Rub around with your hands and arrange in a single layer on baking sheet. Bake for 6-7 minutes until asparagus is firm tender and bright green.
Slice mushroom caps in lengthwise strips. Heat 1 T oil in pan over medium heat. Cook mushrooms until they've released their liquid and are starting to brown. Remove from pan.
Slice zucchini in thin lengthwise strips.
Thinly slice cheeses.
Lower oven to 350F.
Assemble sandwiches: on large baking sheet (can use the same sheet you used for the asparagus), lay out half the bread slices. Lay one slice of cheese on each and top with slices of zucchini, mushrooms, and asparagus (I cut mine in half to fit). Top with another slice of cheese, a slice of bread, and more cheese on top. Place in oven until cheese is slightly melted, about 5-7 minutes.
While sandwiches are heating, fry eggs in same pan you used for the mushrooms.
Reheat sauce until very hot.
TO SERVE: Place one sandwich on a plate Place an egg on top. Pour hot sauce over the entire sandwich until it pools a bit on the sides. The hot sauce will melt the cheese a bit more. Surround with French fries.