Place 4 cups all-purpose flour and 4 teaspoons kosher salt in a large bowl and whisk to combine.
Place 1 1/2 cups warm water and 1 tablespoon barley malt syrup or honey in a small bowl and whisk to combine. Sprinkle 1 packet active dry yeast over the top. Let stand until the yeast dissolves and is foamy, about 5 minutes.
Add the yeast mixture and 1/4 cup of the olive oil to the flour mixture and stir with a flexible spatula until no dry flour remains.
Remove the dough from the bowl. Clean and dry the bowl, then coat the bowl with 1 tablespoon of the olive oil. Return the dough to the bowl and turn to coat in the oil. Cover with plastic wrap and let rise in a warm place until doubled in bulk, about 1 hour. (Alternatively, let rise in the refrigerator for 12 to 24 hours.)
Place 2 tablespoons of the olive oil in a 9x13-inch metal baking pan and rub it into the bottom and sides of the pan. Place the dough in the pan. Gently stretch to the edges of the pan; if the dough is very sticky, use damp hands or a plastic dough scraper.
Cover with plastic wrap and let rise in a warm place until puffed and slightly risen, about 1 1/2 hours. The dough is ready when it appears puffed and bubbly, and it springs back slowly when poked, leaving a small indentation.
Arrange a rack in the middle of the oven and heat the oven to 450℉. Meanwhile, pick the leaves from 2 fresh rosemary sprigs and coarsely chop until you have 1 tablespoon. Mince 3 garlic cloves.
Uncover the dough and sprinkle with the rosemary, garlic, and 1/2 teaspoon flaky salt. Use your fingertips to gently dimple the surface of the dough, then drizzle the remaining 2 tablespoons olive oil evenly over the top.
Bake until the edges of the focaccia are dark-golden brown, 15 to 20 minutes. Garnish with more chopped fresh rosemary leaves if desired.