Place towel or shelf liner beneath stand mixer to prevent wobbling. Mix flour, water, and table salt in bowl of stand mixer fitted with dough hook on low speed until no patches of dry flour remain, 3 to 4 minutes, occasionally scraping down sides and bottom of bowl. Turn off mixer and let dough rest 20 minutes.

Sprinkle yeast and sugar over dough. Knead on low speed until fully combined, 1 to 2 minutes, occasionally scraping down sides and bottom of bowl. Increase mixer speed to high and knead until dough is glossy, smooth, and pulls away from sides of bowl, 6 to 10 minutes.

Using fingers, coat large bowl with 1 tablespoon oil, rubbing excess oil from fingers onto blade of rubber spatula. Using oiled spatula, transfer dough to bowl and pour 1 tablespoon oil over top. Flip dough over once so it is well coated with oil; cover tightly with plastic wrap. Let dough rise at room temperature until nearly tripled in volume and large bubbles have formed, 2 to 2 ½ hours.

Heat butter and oil in 12-inch nonstick skillet over high heat; when foam subsides, stir in salt and sugar. Add onions and stir to coat; cook, stirring occasionally, until onions begin to soften and release some moisture, about 5 minutes. Reduce heat to medium and cook, stirring frequently, until onions are deeply browned and slightly sticky, about 40 minutes longer. Off heat, stir in water; season to taste with pepper. Transfer to large plate and cool to room temperature.

One hour before baking pizza, adjust oven rack to middle position, place pizza stone on rack, and heat oven to 450 degrees.

Coat rimmed baking sheet with remaining 2 tablespoons oil. Using rubber spatula, turn dough out onto baking sheet along with any oil in bowl. Using fingertips, press dough out toward edges of pan, taking care not to tear it. Let dough rest in pan until slightly bubbly, 5 to 10 minutes. Using dinner fork, poke surface of dough 30 to 40 times and sprinkle with kosher salt.

Bake until spotty brown, 15 to 17 minutes. Remove pizza from oven, spread onions evenly over surface, and sprinkle with Gruyère and thyme. Return pizza to oven and continue to bake until cheese begins to brown in spots, 5 to 10 minutes longer.