For the dough, combine the water, yeast, honey, and olive oil in the bowl of an electric mixer fitted with a dough hook. (If the bowl is cold, start with warmer water so it's at least 100 degrees F when you add the yeast.) Add 3 cups of the flour, then the salt, and mix on medium-low speed. While mixing, add 1 more cup of flour, or just enough to make a soft dough. Mix the dough on medium-low speed for about 10 minutes, until smooth, sprinkling it with flour to keep it from sticking to the bowl. When the dough is ready, turn it out onto a floured board and knead it by hand a dozen times. It should be smooth and elastic. Place the dough in a well-oiled bowl and turn it to cover lightly with oil. Cover the bowl with a damp kitchen towel. Allow to rest at room temperature for 30 minutes.
Preheat the oven to 450 degrees F.
Divide the dough into 2 equal parts, rolling each 1 into a smooth ball. If you're only making 1 pissaladiere, place 1 ball on a baking sheet and cover it loosely with a damp towel. Allow the dough to rest for 10 minutes. (If you're not using the other dough, wrap it well and refrigerate or freeze it for the next time.) Roll the dough lightly with a rolling pin, then stretch it to a 10 by 15-inch rectangle and place it on a baking sheet sprinkled with cornmeal.
Spread the tomato paste onto the dough, leaving a 3/4-inch border all around. Artfully arrange the sliced tomatoes and sliced mozzarella on top. Sprinkle with the Cheddar, pepper, and thyme. Brush the edge of the dough with olive oil, and bake for 15 minutes, or until the crust is crisp. Serve hot on a cutting board.