Make dough: In a mixing basin, dissolve yeast in warm water and let foam for 5 minutes. Mix 1 cup flour with 1 tablespoon oil, sugar, and salt until smooth. Slowly add remaining flour until mixture is smooth and workable.
Turn dough onto a lightly floured surface. Knead dough for 5 minutes until elastic. Put dough in a large bowl with 1 teaspoon olive oil and flip to coat. Leave covered for 40 minutes to practically double in size.
Meanwhile, create filling: In a medium bowl, mix cheddar, ricotta, pepperoni, mushrooms, and basil. Mix thoroughly, cover, and chill.
Pre-heat the oven to 375°F. Grease baking sheet.
Pinch dough and divide into two equal pieces. Roll each piece into 1/8-inch circles on a lightly floured surface. Half-fill each circle with cheese and meat.
Fold dough over filling to form semicircle. Seal edges using fork tines. Top each calzone with beaten egg and set on the baking pan.
Bake for 30 minutes in the preheated oven until golden brown. Serve hot.