Mix your yeast, sugar, and warm water in a small bowl and let it sit for about 5 minutes until it gets foamy. In a larger bowl, whisk together the flour and salt, then pour in the yeast mixture and olive oil. Stir everything until shaggy dough forms — don't worry if it looks rough right now.
Turn the dough onto a lightly floured surface and knead for about 8 minutes until it becomes smooth and elastic. The dough should spring back when you poke it, but don't overwork it or it'll get tough. Place it in an oiled bowl, cover with a damp cloth, and let it rise for 1 hour until doubled in size.
Once your dough has risen, punch it down gently and transfer it to a baking sheet lined with parchment paper. Stretch and shape it into a flat rectangle about 1/4-inch thick — your hands will be your best tool here. Don't stress if it's not perfectly shaped because rustic flatbreads are supposed to look a little irregular anyway.
Preheat your oven to 425°F while your dough rises for a second 20-minute rise on the baking sheet. This second rise is crucial for getting that fluffy texture, so don't skip it even though you're impatient. You'll know it's ready when the dough looks puffy and holds an indent when you gently press it.
Brush the flatbread dough lightly with a little of that extra-virgin olive oil and bake it for about 8 minutes until it just starts to turn golden. Don't pull it out too early or the base won't be crispy enough — this is where I learned the hard way after pulling mine out and it was still doughy. Watch for that color change and you'll know you're good.
Remove the flatbread from the oven and immediately arrange your mozzarella slices and tomato halves on top in a pattern that makes you happy. Drizzle with the remaining olive oil and that balsamic glaze, then sprinkle with the sea salt. Return it to the oven for just 5 minutes until the cheese gets all melty and soft.
Pull your caprese flatbread balsamic from the oven and scatter the fresh basil on top while it's still warm — the heat brings out the best basil flavor. Let it cool for about 2 minutes so it won't burn your mouth, then cut it into 8 squares or however many pieces you want. Serve it warm and watch it disappear faster than you'd expect.