Preheat the oven with a pizza stone to 500 degrees F for 30 minutes.
Place a large saute pan over high heat. Add the olive oil and when it is hot, saute the artichoke hearts and mushrooms. Season, to taste, with salt and pepper. Pour off any excess oil and let the vegetables cool.
Roll or stretch the dough, 1/4 at a time, into a 9-inch circle. Place the dough, one at a time, on a lightly floured wooden peel. Brush the circles to within 1-inch of the edge with chili oil. For each calzone, put 1/4 of both cheeses on half of the dough, still leaving the 1-inch border. Top with 1/4 of the artichokes and mushrooms, the garlic and the thyme. Assemble the remaining calzones in the same way.
Moisten the edges of the circles with water. Fold the un-topped half of the dough, trapping as much air inside as possible, over the filling and press the edges together firmly to seal. With the backside of a fork, press the edges of the dough to crimp them.
Slide the calzones onto the pizza stone and bake about 12 minutes, or until the crust is golden brown. Brush the calzones with melted butter and sprinkle them with Parmesan.
In a small bowl, dissolve the yeast and honey in 1/4-cup warm water.
In a mixer fitted with a dough hook, combine the flour and the salt. Add the oil, the yeast mixture, and the remaining 3/4 cup of water and mix on low speed until the dough comes cleanly away from the sides of the bowl and clusters around the dough hook, about 5 minutes. (The pizza dough can also be made in a food processor fitted with the steel blade. Pulse once or twice, add the remaining ingredients, and process until the dough begins to form a ball.)
Turn the dough out onto a clean work surface and knead by hand 2 or 3 minutes longer. The dough should be smooth and firm. Cover the dough with a clean, damp towel and let it rise in a warm spot for about 30 minutes. (When ready, the dough will stretch as it is lightly pulled).
Divide the dough into 4 balls, about 6 ounces each. Work each ball by pulling down the sides and tucking under the bottom of the ball. Repeat 4 or 5 times. Then on a smooth, unfloured surface, roll the ball under the palm of your hand until the top of the dough is smooth and firm, about 1 minute. Cover the dough with a damp towel and let rest 15 to 20 minutes. At this point, the balls can be wrapped in plastic and refrigerated for up to 2 days. Yield: 4 (8-inch) pizzas