Set a 16-by-14-inch pizza stone or baking tiles on the bottom shelf of the oven and preheat to 500° for at least 30 minutes. Meanwhile, in a large skillet, heat 2 tablespoons of the oil. Add the shiitakes and cook over moderately high heat, stirring occasionally, until beginning to brown, about 4 minutes. Add the garlic and cook, stirring often, until lightly browned, about 3 minutes. Transfer the mushrooms and garlic to a plate.
In the same skillet, heat 1 tablespoon of oil. Add the dandelion greens and 1 tablespoon of the water and cook, stirring, until wilted, about 4 minutes. Return the mushrooms and garlic to the skillet, add the remaining 1 tablespoon of water and cook until the liquid is absorbed and the greens are al dente. Season with salt and pepper and transfer to a plate.
On a lightly floured surface, roll or stretch the dough to form two 10-inch rounds. Assemble 1 calzone at a time: transfer 1 dough round to a well-floured rimless cookie sheet. Sprinkle one-quarter of the Fontina on half of the dough, leaving a 1-inch border. Top with half of the dandelion mixture and another quarter of the Fontina. Fold the dough over to enclose the filling and press the edges together; crimp to seal. Using a toothpick, poke a hole in the top. Brush the calzone with olive oil and sprinkle with 1 tablespoon of the Parmesan. Repeat with the remaining ingredients on a second rimless cookie sheet.
Jiggle the cookie sheets to release the calzones and slide them onto the pizza stone. Bake for about 11 minutes, or until the calzones are crisp and the filling is bubbling through the holes. Transfer to a rack. Cut them in half and serve hot.
Make Ahead: The dandelion mixture can be cooked up to 1 day ahead and refrigerated.
Wine Recommendation: A straightforward, medium-bodied Bordeaux, such as the 1995 Michel Lynch Merlot, or a deeper Médoc, such as the 1994 Château de Pez, would offer a pleasantly astringent contrast.