Preheat the oven to 400°F.
In a large ovenproof skillet, heat the butter over medium-high heat. Toss in the onion and sauté until softened, 4 to 5 minutes. Add the garlic and sauté until it smells fragrant, about 1 minute.
Toss in the squash and potatoes and stir until coated with the onion mixture. Season with salt and stir in the chile paste (if using). Add the mixed vegetables and cream and allow everything to come to a simmer. Stir in half of each of the grated cheeses.
Transfer the skillet to the oven and bake until the potatoes and squash are tender, about 25 minutes—to check, just poke them with a knife or skewer.
Remove the skillet from the oven and turn the broiler to high.
Top the potatoes and vegetables with the remaining grated cheeses and broil until the cheese is bubbling and browned in spots, about 8 minutes. Serve hot!