Position a rack in the center of the oven, set a pizza stone on the rack, and heat the oven to 450°F. (If you don’t have a pizza stone, set a rimmed baking sheet upside down on the rack to serve as a baking platform.) Let the pizza dough come to room temperature.
Meanwhile, heat 1 Tbs. of the olive oil in a 12-inch skillet over medium-high heat until shimmering hot. Add the onions and a generous pinch of salt. Cook, stirring, until the onions are golden and softened, about 10 minutes. Make a space in the skillet and add the minced garlic. Let sizzle until fragrant, then stir into the onions and cook, stirring occasionally, to blend the flavors, about 2 minutes. Season to taste with salt. Remove from the heat.
In a 1-quart saucepan over low heat, heat the remaining 3 Tbs. oil with the anchovies and thyme. Cook, stirring and breaking up the anchovies with a wooden spatula until they’re dissolved, about 1-1/2 minutes. Remove from the heat.
Cut the pizza dough in half. Working with half at a time on a lightly floured surface, use your fingers to flatten and stretch it. Then, using a lightly floured rolling pin, roll the dough into a 9x13-inch rectangle (rounded corners are fine). If at any time the dough begins to shrink back, let it rest for 10 minutes and then roll again. Transfer the dough to a piece of parchment. Brush the dough with half of the anchovy oil, then spread with half of the onion mixture. Sprinkle half of the capers and olives evenly on top. Slide the parchment and dough onto a pizza peel (or an inverted baking sheet), then slide the parchment and dough onto the hot pizza stone (or baking sheet). Bake until golden-brown, 10 to 12 minutes. Prepare the second tart while the first cooks.
Remove the first tart from the oven and transfer to a cutting board. Bake the second tart. Let the tarts cool for at least 5 minutes before slicing and serving.