To make the dough,in a stand mixer with the dough hook, mix the flour, sugar, table salt and yeast on medium until well combined, about 15 seconds. With the mixer on low, drizzle in the oil followed by the water. Scrape the bowl. Mix on medium, occasionally scraping the bowl and dough hook, until the mixture forms a shaggy, sticky dough, 4 to 6 minutes. Scrape the sides of the bowl and gather the dough in the center, cover with plastic wrap and let rise at room temperature until doubled, 1½ to 2 hours.

Meanwhile, prepare the toppings. In a 12-inch skillet over medium, combine 2 tablespoons oil, the onions, thyme sprigs, bay and 1 teaspoon kosher salt. Cook, stirring occasionally, until the onions are lightly golden, 30 to 35 minutes. Stir in 1 teaspoon of the chopped anchovies; cook, stirring occasionally, until the onions are golden brown and fully softened, another 5 to 10 minutes. Remove from the heat, then discard the thyme sprigs and bay. Stir in ½ teaspoon pepper, the minced thyme and vinegar; set aside.

Brush the bottom and sides of a 13-by-18-inch rimmed baking sheet with the remaining 2 tablespoons oil, then sprinkle evenly with the semolina. Turn the dough onto the baking sheet, then, using oiled hands, press and stretch it into an even layer that covers the surface of the baking sheet; it’s fine if the dough doesn’t completely fill the corners. Cover with plastic wrap and let rise at room temperature until puffed and almost doubled, about 30 minutes. Meanwhile, heat the oven to 450°F with a rack in the middle position.

Distribute the onions evenly over the dough, leaving a ½-inch border around the edge. Scatter the remaining chopped anchovies over the onions, followed by the olives. Bake until the edges of the dough are golden brown, 20 to 25 minutes. Using a large metal spatula, slide the pissaladière onto a cutting board and sprinkle with the parsley. Cool for at least 10 minutes. Serve warm or at room temperature.