Mix the bread flour, whole wheat flour, yeast, sugar, and salt in the large mixing bowl. Make a well in the flour and add the warm water and olive oil. Mix and knead thoroughly until the dough clings together and cleans the sides of the bowl.
Turn the dough out onto a floured work surface and knead for about 5 minutes, until the dough is smooth and soft. Dust with additional flour if the dough is sticking to your hands.
Spread some olive oil around the other large mixing bowl and lay the dough in the bottom, turning it over so that the entire surface is coated with a light layer of olive oil. Cover the bowl with plastic wrap and a kitchen towel. Set the bowl in a warm place to rise until doubled, about 90 minutes.
While the dough is rising, combine all of the sauce ingredients in the 2-quart bowl and stir. Set aside.
Preheat the oven to 500° F/260° C.
Punch down the dough and knead briefly. Divide into 6 equal pieces and shape each into a round about 6 inches across. For each pot pie, coat the inside of the ramekin liberally with olive oil (if there is not enough olive oil the cheese will stick during turnout), arrange three mozzarella slices in the bottom and around the sides of the ramekin making sure they overlap. Add mushrooms and meatballs (or other fillings) as desired.
Ladle the sauce over the fillings just until the ramekin is almost full. Rub the outside of the ramekin with olive oil, then stretch a round of dough across the top and down the sides to nearly the bottom and brush the top with olive oil.
Arrange the pies on a foil-lined baking sheet and bake in the preheated oven until they are toasted brown on top, about 15 to 20 minutes. Remove from the oven and transfer ramekins to a wood cutting board or other heat resistant surface. Place a serving plate on top of each pie, then invert it.
Turn the bottom of the ramekin with an oven mitt. Lightly run a knife or spoon between the outside of the ramekin and the inside of the dough bowl being careful not to slice through the bottom. Gently lift the ramekin out, turning it as you go, and scoop out the cheese if necessary. Use a fork and knife to eat and enjoy.