Make the dough: In a small bowl, dissolve the yeast and honey in 1/4 cup of the warm water. In a stand mixer fitted with a dough hook, combine the flour and salt. Pour in the oil and, when absorbed, scrape in the dissolved yeast. Add the remaining 1/2 cup water and knead on low speed about 5 minutes.
Turn the dough out onto a floured board and knead 2 or 3 minutes longer. The dough should be smooth and firm. Cover with a damp towel and let rise in a warm spot for 30 minutes. (Dough will stretch when lightly pulled.)
Divide the dough into 4 balls, about 6 ounces each. Work each ball by pulling down on 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 dough is smooth and firm, about 1 minute. Cover with a damp towel and let them rest for 15 to 20 minutes. At this point, the balls can be loosely covered with plastic wrap and refrigerated for 1 to 2 days.
Place a pizza stone in the oven and preheat it to 450 degrees F.
Heat a large skillet over high heat and cook the eggplant and artichoke hearts separately in 1 tablespoon olive oil, adding more oil if necessary.
To prepare each pizza, place a ball of dough on a lightly floured surface. Press down on the center, spreading the dough into a 7 or 8-inch circle, with the outer border a little thicker than the inner circle. (Or roll out with a rolling pin.)
Place dough on a pizza paddle or a cutting board and brush it with olive oil. Arrange the mozzarella and fontina cheeses over the inner circle. Top with the eggplant and artichoke hearts. Add a layer of tomato slices and sprinkle over the Parmesan and thyme. Slide the pizza onto the pizza stone and cook until the dough is brown and the cheese is bubbling, about 8 to 10 minutes.
Remove carefully, cut into 6 slices, and sprinkle with basil.