Dissolve yeast in warm water in a mixing bowl; let sit until foamy, about 5 minutes. Add 1 tablespoon olive oil, sugar, and salt; mix in 1 cup flour until smooth. Gradually stir in remaining flour until dough is smooth and workable.

Turn dough out onto a lightly floured surface. Knead dough until it is elastic, about 5 minutes. Place 1 teaspoon olive oil in a large bowl; lay dough in the bowl and flip to coat with oil. Cover and let rise until almost doubled in size, about 40 minutes.

Meanwhile, make the filling: Combine Cheddar cheese, ricotta cheese, pepperoni, mushrooms, and basil leaves in a medium bowl. Mix well, cover, and refrigerate to chill.

Preheat the oven to 375°F (190°C). Grease a baking sheet.

Punch down dough; divide into 2 equal pieces. Roll each piece out on a lightly floured surface into 1/8-inch-thick circles. Fill each circle with 1/2 of the cheese and meat filling.

Fold dough over filling to make a semicircle. Press edges down with the tines of a fork to seal. Brush the top of each calzone with beaten egg and place on the prepared baking sheet.