Peel and chop the onions. In a large frying pan over medium-high heat, heat the olive oil and add the onions and 1 teaspoon of the salt. Cook, stirring once in a while, until the onions soften completely, about 8 minutes. While the onions cook, peel and mince the garlic. Add it to the onion and cook another 2 minutes. Stir in the thyme, rosemary, and pepper; take off the heat. Divide the mixture, transferring half to a small bowl to set aside and half to the bowl in which you’ll make the dough.

Add 1 1/2 cups warm water to the bowl for the dough along with the milk, the remaining 2 teaspoons of the salt, and the sugar. Stir to combine. Remove about 1/2 cup of the liquid to a small bowl, sprinkle in the yeast, and let sit until foamy, about 5 minutes. Stir into the onion-milk mixture. Stir in the flour 1 cup at a time. Turn the dough out onto a floured surface and knead until a smooth and elastic dough forms, at least 5 minutes.

Grease a large bowl, put the dough in the bowl, cover with a clean kitchen towel, and let sit until doubled in bulk, about 2 hours.

Punch down the dough and let it rise until doubled in bulk again, about 1 hour.

Generously coat a large rimmed baking sheet (11x17) with olive oil. Punch for the dough and turn it onto the oiled sheet. Gently pull and push the dough so it more or less fills the baking sheet. Let rise, uncovered, until pillowy looking, about 1 hour.

Preheat an oven to 450°F. Use your hands to gently spread the reserved onion mixture over the dough. Your fingers should be oily from that and all set to poke holes: poke deep holes all over the dough. Sprinkle the dough with the coarse salt and bake until browned all over and crisp along the sides, about 30 minutes. Transfer the focaccia to a cooling rack before serving.