Preheat oven to 425 degrees. Brush a rimmed baking sheet with oil. On a lightly floured work surface, roll dough into a 16-inch round. Spread mushrooms on bottom half of dough, leaving a 1-inch border. Top mushrooms with artichoke hearts and fontina. Fold top half over and roll and pinch edges to seal. Brush off excess flour.
Carefully transfer calzone to baking sheet. Lightly brush with oil, season with salt, and bake until golden brown, 22 to 25 minutes.