Melt butter in a large skillet over medium-high heat. Add onions and mushrooms; sauté until vegetables are tender and mushrooms give up some of their liquid. Stir in garlic, artichokes, spinach, and basil. Cook an additional 1 to 2 minutes. Remove from heat and cool to room temperature.
Add ricotta and parmesan cheeses to cooled vegetables; mix together well.
Divide pizza dough into 10 pieces. Roll each piece into an 8-inch circle. Place about 1/3 cup of filling on pizza dough. Fold in half and pinch edges together. Place on a baking sheet and brush with olive oil, if desired.
Bake calzones for 20 minutes in 400°F oven. Eat hot or cold.