Heat the oil in a large pot over medium-high heat. Add the onions and cook, stirring, until softened and starting to brown, about 5 minutes. Stir in the carrots, peppers and courgette and cook, stirring, until softened, about 15 minutes. Add the garlic, 1 teaspoon salt and 1/4 teaspoon pepper. Continue to cook, stirring, until the vegetables are lightly browned, for 10 minutes.
Stir in the crushed tomatoes. Add 1 cup water to the empty tomato can and pour into the pot. Reduce the heat to medium low and simmer until the vegetables are very soft, the sauce has thickened slightly and the rawness from the canned tomatoes has mellowed, about 20 minutes. Puree the vegetables with a hand blender or transfer to a blender and puree until smooth, holding down the lid with a kitchen towel to prevent the hot sauce from spraying. Season with additional salt and pepper if needed and cool. You will have about 5 cups of sauce.
While the sauce simmers, lay the English muffin halves in a single layer on a baking sheet and lightly toast in a toaster oven set to broil or under an oven broiler, 2 to 3 minutes. Spread the top of each muffin with 1 to 2 tablespoons of sauce. Combine the 2 cheeses and sprinkle over the sauce. To serve as a snack right away, add your desired toppings and place the assembled pizzas under the grill or in the oven until the cheese is melted, 2 to 3 minutes.
Cook's Notes: The sauce recipe makes more than you need for 12 pizzas. Keep the extra in the refrigerator for up to 1 week and use in any recipe where you would normally use marinara sauce; or freeze in 1-cup amounts in plastic freezer storage bags for your next batch of English muffin pizzas.
To make ahead: Place the entire tray of mini pizzas in the freezer and freeze until hard, about 1 hour. Stack in a plastic freezer bag until ready to eat. To reheat, place the frozen pizzas on a piece of heavy-duty foil, add any toppings you like and bake in the oven until golden, hot and the cheese is melted.