Measure the water in a measuring cup, stir in the yeast and let develop for 10 minutes. It should look foamy when it's ready.
Meanwhile, add the flour and kosher salt to the bowl of a stand mixer fitted with a dough hook. Turn on to whisk together and run for a minute or so to create a well. With the mixer running, add the water and yeast in a stream. Turn the mixer up to speed 2 and knead until the dough comes away from the bowl and it looks smooth and feels elastic but still sticky, 5 to 7 minutes. Flour your hands and knead the dough for a minute on the counter.
Grease a large glass bowl with 2 tablespoons olive oil. Add the dough, flip on both sides and cover with a clean tea towel. Place the dough in a warm spot to rise until it is double in size, about 1 hour.
Spread a half sheet tray with the remaining olive oil. Lay the dough on top and stretch and pat out until it reaches 10-by-16-inches, an oval rough shape is ideal. Use your fingertips to press in small indentations. Sprinkle evenly with the cherry tomatoes, sea salt and rosemary leaves. Let rise again for 20 minutes.
Place on the middle rack of the oven and bake until puffed and golden, 20 to 25 minutes. Slice into wedges or squares for serving and drizzle with more olive oil, if desired.