In a medium bowl, proof yeast by mixing the yeast and sugar with warm water. Let sit for 8-10 minutes or until foamy. Add oil and salt and mix in flour 1 cup at a time until dough is smooth.
Turn out dough onto a lightly floured surface and knead for about 5 minutes, or until elastic. Add a teaspoon of olive oil to a large bowl and flip dough to coat with oil. Cover and let rise for about 45 minutes, or until doubled in size.
While the dough is rising, make the filling by combining mozzarella cheese, ricotta cheese, pepperoni, and basil in a medium bowl. Refrigerate until ready to add to dough.
Preheat oven to 375°F. Grease a baking sheet and set aside.
Once the dough has risen, punch it down and divide it into 2 equal pieces. Divide the pieces into 3-4 balls. Roll each ball out onto a lightly floured surface into ⅛-inch thick circles.
Add ¼ of the filling and fold the dough over to make a semi-circle. Press down curved edges with the tip of a fork to seal. Brush tops with beaten egg and place on the prepared baking sheet. Use a sharp blade to cut a "v" shape venting slit on top of each calzone.
Bake for 24-28 minutes, based on the size of the calzones, or until golden brown. Serve warm.