Step 1In a bowl of a stand mixer or large mixing bowl, dissolve the yeast into warm water along with 1 tablespoon all-purpose flour and one-half teaspoon sugar. Set aside until the yeast begins to activate and become foamy, several minutes.
Step 2Meanwhile, in a separate large bowl, thoroughly whisk together the remaining 2 cups of all-purpose flour with the bread flour and salt.
Step 3Add the room-temperature water and oil to the yeast mixture, and whisk to combine.
Step 4With the mixer running or by hand, add the dry ingredients to the yeast mixture until combined. Knead with the paddle attachment on medium speed, or by hand, until smooth, about 10 minutes. Form the dough into a ball and place the ball in a separate, large, well-oiled bowl. Smooth olive oil over the ball and cover the bowl loosely with plastic wrap. Refrigerate the dough overnight.
Step 5Remove the dough from the refrigerator and divide into 8 equal pieces, each about 3.7 ounces in weight. Form each piece into a ball and place on a well-oiled baking sheet (you will fit about 4 per sheet, using 2 sheets total). Set the sheets aside in a warm place until the dough has puffed and doubled in size, about 1 hour.
Step 6Dimple the dough with your fingers (oil your fingers so the dough does not stick), then set aside until puffed again, about 30 minutes. Dimple again, then repeat once more. Heat the oven to 450 degrees.
Step 7After the final dimpling, add the fruit to the focaccia, distributing the fruit evenly among the pieces and pressing the fruit firmly into each focaccia. Sprinkle the chopped rosemary evenly over the focaccia, and drizzle olive oil over each piece. Sprinkle the remaining half-cup sugar over the focaccia, about 1 tablespoon per piece (try not to get too much sugar on the sheets as it can get sticky and make it difficult to remove the focaccia).
Step 8Place the sheets in the oven and bake until puffed and golden-brown around the edges, about 15 to 20 minutes, rotating the sheets halfway through for even baking. Remove and cool slightly before removing from the sheets and serving.