To start, combine the semolina flour and strong white bread flour in a large bowl. Make a well in the center. Sprinkle the yeast and sugar into the well. Pour in the lukewarm water, mixing slowly until combined. Knead the dough on a floured surface for about 10 minutes until it becomes smooth.
Grease a bowl with olive oil, place the dough inside, and cover it with a clean cloth. Let it rise in a warm area until it's doubled in size, approximately 1-2 hours. This is the moment of magic!
Once risen, punch down the dough to release air. Lightly flour your surface again and stretch out the dough into a rectangle, about 1-inch thick. Transfer it to a greased baking sheet.
Using your fingers, press down into the dough to create dimples. This is not just for show; it helps hold the toppings. Cover it again and let it rise for another 30 minutes.
While your dough is resting, let’s prepare the toppings. For the balsamic onion topping, heat a pan over medium heat. Add the sliced onions and cook until soft, about 10 minutes. Add the balsamic vinegar, thyme, and honey (if using), and simmer for about 5 minutes.
For the basil and cherry tomato topping, mix the basil, tomatoes, garlic, and white wine vinegar in a bowl. If you're feeling fancy, splash in some olive oil.
Preheat your oven to 220°C (428°F).
Once your dough has risen again, drizzle it lightly with extra virgin olive oil. Spread the onion mixture evenly on one side, the tomato mix on the other, and sprinkle cheese on the entire surface. Finish with rosemary and cracked pepper.
Put the focaccia in the oven and bake for 25-30 minutes. You'll know it's done when it’s golden brown and the house smells divine!
Let the focaccia cool on a wire rack for a few minutes. Drizzle a bit more olive oil before slicing. Now, cut into squares and prepare for oohs and aahs!