Coat a large bowl with 3 Tbsp. oil. In another large bowl, whisk flour, kosher salt, and yeast to combine. Add water and mix with a silicone spatula just until a wet dough forms. Transfer to prepared bowl. Cover with plastic wrap and let sit at room temperature until risen and doubled in size, about 1 hour.

Meanwhile, in a large stainless steel or cast-iron skillet over medium heat, heat 1 Tbsp. oil; swirl to coat pan. Add sausage and cook, breaking up into small pieces, until starting to brown on the outside, 10 to 12 minutes. Transfer sausage to a plate; reserve remaining oil in pan.

Arrange a rack in center of oven; preheat to 450°. Grease a 13" x 9" metal baking pan with remaining 2 Tbsp. oil and reserved 1 Tbsp. sausage oil from skillet.

Transfer half of dough to prepared pan. Using your fingertips, spread as much as possible to each corner. Top dough with half of sausage crumbles and half of cheese. Place remaining dough on top. Using your fingertips, gently spread remaining dough over filling. Top with remaining sausage and cheese, then tomatoes and rosemary leaves.

Create 6 indentations in dough. Crack an egg into each; season with sea salt and pepper.

Bake breakfast focaccia, rotating pan halfway through and checking at around 20 minutes, until dough is puffed and golden brown, and egg whites are set but yolks are still slightly runny, 23 to 28 minutes. If you like your eggs more set, leave it in closer to 28 minutes. Let rest 5 minutes, then serve warm.