Finely chop 1 onion. In a large frying pan over medium-high heat, add 3 tbsp. olive oil, the chopped onion, garlic, and 1/2 tsp. salt. Cook, stirring occasionally, until onions are soft, about 3 minutes. Set aside.
Meanwhile, in the bowl of a stand mixer, dissolve yeast in 1 cup warm water (90° to 105°). Let sit until foamy, about 5 minutes.
Attach dough hook and, with mixer on low, add milk, 1 tbsp. salt, the sugar, and onion mixture to yeast. Work in flour, 1 cup at a time, until you have a smooth and elastic dough, about 3 minutes.
Transfer dough to a large, oiled bowl. Cover with a clean towel or plastic wrap and let sit until doubled in bulk, about 1 1/2 hours. Punch down, cover, and let sit until doubled in bulk again, about 1 hour. Oil a large rimmed baking sheet (ideally 12 in. by 16 in.). Punch down dough again, transfer it to baking sheet, and use your hands to spread and gently push it into an even layer. Cover and let sit until soft and puffy, about 45 minutes.
Meanwhile, halve and thinly slice remaining 2 onions. In a large frying pan over medium-high heat, add remaining 2 tbsp. olive oil, the onions, and 1/2 tsp. salt. Cook, stirring, until onions are soft and beginning to turn translucent, about 3 minutes. Set aside.
Preheat oven to 450°. Using your fingers, poke holes straight down into the risen dough. Spread cooked onion mixture evenly over top and sprinkle with remaining 1/2 tsp. salt. Bake until golden, 30 to 35 minutes. Cool in pan on a wire rack. Serve warm or at room temperature.
Note: Nutritional analysis is per 4-in. square.