Combine the water, yeast, sugar, herbes de Provence and the 2 Tbsp. olive oil in a medium bowl, or in the bowl of your stand mixer fitted with a dough hook.
If using a stand mixer, add the flour and mix on medium speed until a smooth dough forms that pulls away from the sides of the bowl. Continue mixing and kneading the dough 5 minutes.
If mixing the dough by hand, add 1 1/4 cups of the flour into the yeast mixture. Mix with a spoon until the dough loosely clumps together. Transfer the dough to a work surface, scraping the sides of the bowl if necessary. Use the remaining 1/4 cup of flour to lightly flour the work surface and top of the dough. Knead six to eight minutes, until you have smooth, but still slightly sticky dough.
Once kneaded, place the dough in a deep, lightly oiled bowl. Cover with plastic wrap and let rise at warm room temperature until doubled in size, about 60 minutes.
While the dough rises, heat the 2 tsp. oil in a skillet set over medium heat. Add the onions and cook until softened, about 5 minutes. Remove from the heat and cool.
When the dough has doubled in size, lightly oil a nonstick baking sheet. Set the dough on the baking sheet. Gently press and stretch the dough into a thin oblong about 10” long and 9” wide. Spread the top of the dough with the onions. Sprinkle the onions with the salt. Let formed foccacia bread rise 30 minutes.
Preheat the oven to 425 degrees F. When the bread has risen 30 minutes, bake 20 to 23 minutes, or until puffed and golden.