In a large nonstick skillet, melt 2 tablespoons of the butter in 2 tablespoons of the olive oil. Add the mushrooms and cook over high heat, stirring occasionally, until browned and tender, about 5 minutes. Season with salt and pepper and transfer to a medium bowl.
In the same skillet, melt the remaining 1 tablespoon of butter in the remaining 1 tablespoon of olive oil. Add the kale and a generous pinch of salt and cook over moderately high heat, stirring occasionally, until just tender, about 5 minutes. Season with salt and pepper.
In a small bowl, mix the 1 cup of tomato sauce with the olive tapenade and basil. In another small bowl, toss the Fontina with the mozzarella.
Preheat the oven to 450° and lightly brush 2 large rimmed baking sheets with olive oil. Cut the pizza dough into 18 pieces and roll into balls. On a lightly floured work surface, roll out 1 ball to a 5-inch round, a scant 1/8 inch thick. Spoon 1 tablespoon of the olive-tomato sauce on one half of the round, then top with 1 tablespoon each of the mushrooms, kale and cheese. Fold the dough over to form a half moon and press the edge to seal tightly. Crimp the edge with a fork or pinch at intervals to make pleats. Transfer the calzone to a baking sheet and brush lightly with olive oil. Repeat with the remaining dough and fillings.
Bake the calzones in the upper and lower thirds of the oven for 12 to 14 minutes, shifting the pans from top to bottom and front to back halfway through baking, until golden and puffed. Transfer to a platter and serve hot, with tomato sauce for dipping.
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