Make the Tart Dough

1. In a food processor, pulse the flour, cornmeal, sugar and salt until just combined. Add the butter and pulse a few times until the mixture resembles coarse, pea-sized crumbs.
2. Combine the egg yolk with 1/4 cup of the ice water. With the food processor running, add the liquid to the dough until it begins to come together. To test, pinch the dough with your fingers, and if it holds without being sticky, it is ready. Add more ice water, a tablespoon at a time, if needed.
3. Pour the dough out onto a lightly-floured work surface and press the dough gently together to form a mound. Divide the dough in half, and shape each half into a disc. Wrap each disc in plastic wrap and chill in the refrigerator for at least an hour.

Make the Galette

1. Preheat the oven to 375 degrees F.
2. On a piece of parchment paper, roll out the dough into a circle about 1/8-inch thick. In order to keep the dough properly chilled, work with just half the dough at a time and leave the other half in the fridge. If the dough begins to stick, dust lightly with flour.
3. Divide the cheese between the two tarts, leaving a 2-inch edge all the way around. Layer the eggplant and tomato slices on top of the cheese, overlapping as required. Season with salt and pepper. Fold the edges of the dough up around the filling, pleating as you go. Pinch together any tears.
4. Whisk together the egg and the milk, and brush the exposed edges of the dough with the egg wash. Transfer the parchment paper to a sheet pan, and bake until the crusts are golden brown and the cheese begins to bubble, about 30-40 minutes. Let cool on a baking rack. Serve warm with a sprinkling of Parmesan or fresh thyme.