For the dough, add water to a large bowl or the bowl of a standing electric mixer. Scatter yeast over the water’s surface. Add sugar and stir to combine. Let stand until bubbly, about 5 minutes.
Add 1 cup of the flour, the olive oil, and salt to the bowl and mix with a wooden spoon or the mixer's paddle attachment until well combined. Add remaining flour, about ¼ cup at a time, mixing by hand or with the paddle until the dough clears the sides of the bowl.
If working by hand, turn the dough out on a lightly floured surface and knead, adding flour a little at a time if the dough still feels sticky. Knead until dough is smooth and elastic. If using an electric mixer, switch to the dough hook and knead on medium speed for about 5 minutes until the dough is smooth and elastic. If the dough feels too sticky, add a bit more flour and knead for another minute or two.
Transfer dough to a lightly oiled bowl, turning the dough once to coat the top with oil. Cover loosely and let the dough rise in a warm place until doubled in volume, about an hour.
For the sauce, heat olive oil in a saucepan over medium-low heat. Add garlic and saute until fragrant, about 1 minute. Stir in crushed tomatoes, basil, and oregano. Simmer over low heat until thick, about 30 minutes. Season with salt and pepper to taste.
Preheat oven to 425° F. Lightly brush 12 standard-size muffin cups with oil.
On a lightly floured surface, turn out the dough and divide it into 12 evenly-sized pieces. Shape the pieces into rounds and let them rest for 10 to 15 minutes while preparing the fillings.
Using just enough flour to prevent sticking, roll the rested balls of dough into 4 to 5-inch circles. Tuck the disks into the prepared muffin cups, pressing a thin edge of dough around the outside of each well to hold the dough in place. Gently press the dough to bottom. It's okay if it doesn't adhere to the bottom and sides of the cups as the weight of the fillings will pull it down.
Add 1 tablespoon each of cheese and sauce over the dough in each cup. Top the sauce with about one tablespoon of assorted fillings. Layer an additional tablespoon of sauce and cheese over the fillings.
Bake for 12 to 15 minutes, until the dough turns golden in color and the cheese is melted. Let rest for 2 minutes before removing from the pan. Garnish with fresh basil, if desired, and serve warm. Take care when eating as the filling holds it heat for some time.