Make the pastry dough and allow to chill while the rest of the ingredients are prepared.
Preheat the oven to 300°F, using the convection setting if you have one.
Thinly slice or dice the sweet onions. Melt 2 tablespoons salted butter in a large skillet over medium heat, then add the onions to the pan.
Slowly cook the onions, stirring often, until they soften and begin to brown. Season with salt and pepper. Be careful not to let the onions scorch or burn - caramelizing onions is a slow process. Lower the heat or add a little water or oil to the pan if you find the onions are beginning to burn or stick.
Continue cooking and stirring the onions over the next 30-45 minutes, until a golden brown color is reached and the onions are completely soft.
While the onions cook, prepare the zucchini. Using a knife or mandoline, slice the zucchinis into ¼ inch thick circles. Lightly salt the circles, then place them into a baking dish. Bake the zucchini slices at 300°F for 20-30 minutes until partially cooked. This step will reduce the amount of moisture added to the galette from the zucchinis.
Prepare the Thyme Goat Cheese: Using a spatula, beat together the goat cheese, half and half, thyme leaves, ½ tsp salt, and 1 tsp freshly cracked black pepper until a smooth mixture is formed.
Preheat the oven to 375°F.
Assemble the galette: Prepare a baking sheet by lining it with parchment paper, then sprinkling ½ tablespoon semolina flour onto the parchment paper. Lightly dust a clean counter with flour. Roll out the pastry dough with a floured rolling pin into a circle, about 14 inches in diameter. The dough should be about ⅛ inch thick evenly across the circle. If the dough deeply cracks outward toward the edges, lightly pull, overlap, and pinch the cracks back together. Some small (1-2 inch) cracks toward the edges are fine.
When you are finished rolling, transfer the dough to the prepared baking sheet. I like to do this by gently folding the dough circle in half, then in half again, taking care not to crack or break the dough. Then lift the folded dough onto the baking sheet and unfold back into the original circle.
Sprinkle the remaining ½ tablespoon of semolina flour over the surface of the pastry dough. Spread the goat cheese mixture onto the dough, leaving a 2-3 inch border of dough around the edges. Layer the caramelized onions over the goat cheese in an even layer, then arrange the zucchini circles in an even layer over the onions.
When you have finished covering the onion layer with zucchini circles, sprinkle the zucchini with salt and pepper and drizzle with about 1 tablespoon extra virgin olive oil.
Gently fold the edges of the dough over the zucchini. Beat the egg, then brush the beaten egg over the dough. Sprinkle the dough with salt and pepper. Sprinkle additional thyme leaves over the zucchini, if desired.
Bake the galette at 375°F for 50-55 minutes, until the crust is golden and flaky.
Allow the galette to cool for 5-10 minutes, then slice and serve with a fork.