In a small bowl, add the water and sprinkle the yeast on top of the water. Add 3 tablespoons of the oil and whisk together. Set aside.
In large mixing bowl, add the flour and 3 teaspoons of salt; whisk together or mix together on low speed in your mixer.
Add the yeast mixture and about half of the flour mixture. Stir with a rubber spatula until it is combined. Attach the dough hook to your mixer and add the remaining flour. Mix on low speed for about three minutes. If the dough seems to be too dry, add warm water a teaspoon at a time until you obtain a softer dough.
This is how my dough looked. It is a dough that is elastic and a little moist.
Form the dough into a ball and place into an oiled bowl (when I put the dough in the bowl I swish the dough around the bottom of the bowl and then flip it over so all of the dough is covered in a light film of oil). Cover with plastic wrap and let it rise at room temperature until doubled in size (about 1-1 1/2 hours).
While the dough is rising, caramelize the onion if using it for topping.
In a large skillet, heat 1 tablespoon extra virgin olive- oil over medium-low heat. Add the onions to the skillet and sprinkle 1 teaspoon of sugar over it. Stir occasionally and cook until tender (takes about 10-15 minutes until golden).
This was my dough after 1 hour.
Place the dough on a parchment lined pan. Pat and press the dough gently until the dough fills the pan completely. If the dough resists, let it rest for a few minutes before continuing (sometimes the dough can be moody). Cover the dough with a piece of plastic wrap and let it rise again until doubled in size (about 40-60 minutes).
Set a rack in the lower third of the oven and preheat to 425 F.
ASSEMBLE AND BAKE
When the dough has doubled in size, remove the plastic wrap. Dimple the surface of the risen focaccia using your fingertips.
Brush on the remaining olive oil to the surface of the dough. Sprinkle on the sea salt (add the amount of salt you prefer. We like it a little saltier. You could even add on crushed black pepper if you'd like) and press the cherry tomatoes gently into the dough. Sprinkle on the chopped rosemary.
Bake the focaccia for about 25 minutes, or until golden brown. Lift the focaccia from the pan with the parchment paper edges onto a cutting board. Let cool on a rack. Cut and serve.