Make dough: Dissolve yeast in warm water in a mixing bowl; let sit until foamy, about 5 minutes. Add 1 tablespoon 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.

Make filling: Meanwhile, 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 degrees F (190 degrees 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.

Bake in the preheated oven until golden brown, about 30 minutes. Serve hot.