Step 1

Preheat the broiler. In a large skillet over medium-high heat, sauté the ground beef or turkey until browned, about 7 minutes. Add the onion and sauté 2 minutes. Add the carrot and sauté 2 minutes. Add the squash and sauté 1 minute more.

Step 2

Stir in the tomato paste and 1 1/2 cups water, stirring until the paste has dissolved. Add the garlic, chili powder, paprika, and oregano, and season with the salt and pepper. Reduce heat to medium and continue to cook until the mixture has thickened, 8 to 10 minutes.

Step 3

Divide the cheese among the bottom halves of the hamburger buns. Transfer both halves of the buns to the broiler, open-faced, and toast until the cheese has melted and the top buns are toasted.

Step 4

Remove the buns from the oven and fill each sandwich with the squash-and-meat mixture. Serve immediately.