Prepare the dough:
Coat a large bowl with 3 tablespoons of olive oil. In another large bowl, whisk together the flour, kosher salt, and yeast. Add the lukewarm water and mix with a silicone spatula until a wet dough forms. Transfer the dough to the prepared bowl, cover with plastic wrap, and let sit at room temperature until the dough has risen and doubled in size, about 1 hour.

Cook the sausage:
While the dough is rising, heat 1 tablespoon of olive oil in a large stainless steel or cast-iron skillet over medium heat. Add the ground sausage and cook, breaking it up into small pieces, until it starts to brown, about 10 to 12 minutes. Transfer the cooked sausage to a plate and reserve the remaining oil in the pan.

Prepare the pan:
Preheat the oven to 450°F (230°C). Grease a 13″ x 9″ metal baking pan with the remaining 2 tablespoons of olive oil and the reserved 1 tablespoon of sausage oil.

Assemble the focaccia:
Transfer half of the risen dough to the prepared pan. Using your fingertips, spread the dough as evenly as possible into each corner of the pan. Top the dough with half of the cooked sausage and half of the shredded cheddar. Place the remaining dough on top and gently spread it over the filling. Top with the remaining sausage and cheese, followed by the halved cherry tomatoes and chopped rosemary leaves.

Add the eggs:
Use your fingers to create 6 indentations in the dough. Crack an egg into each indentation and season with flaky sea salt and freshly ground black pepper.

Bake the focaccia:
Bake the focaccia in the preheated oven, rotating the pan halfway through. Check the focaccia around 20 minutes. Bake for 23 to 28 minutes, until the dough is puffed and golden brown and the egg whites are set but the yolks remain slightly runny. If you prefer your eggs more set, leave the focaccia in the oven closer to 28 minutes.