In a large bowl, whisk 1⁄4 teaspoon sugar and the yeast with 1⁄2 cup plus 2 tablespoons lukewarm water and let stand until foamy, about 10 minutes. Stir in the semolina flour, 1 tablespoon olive oil, and the 1⁄2 teaspoon salt until the dough comes together. Scrape the dough onto a lightly floured work surface and knead until smooth, about 8 minutes. Transfer the dough to a clean bowl, cover with plastic wrap, and let stand at room temperature until doubled in size, about 2 hours.

Heat the oven to 450°. Line a 9-by-5-inch loaf pan with parchment paper.

Scrape the dough onto a lightly-floured large work surface. Using a rolling pin, flatten the dough into a 1⁄16-inch-thick, 24-by-18-inch rectangle, and position the rectangle so that a long side is nearest to you.

Lightly score the dough into 18” x 6” quarters; don’t cut all the way through, just mark it. Spread about half the tomato sauce over the middle two quarters of the rectangle, then sprinkle the sauce with half of the caciocavallo slices. Fold the two plain sides of the dough over so their edges slightly overlap in the middle of the rectangle. You’ll now have a rectangle that’s about 18” x 12”.

Lightly score the dough into four 4 1/2" x 12” sections. Spread a scant 2/3 cup of the remaining tomato sauce evenly over the center two sections and top with about two-thirds of the remaining cheese. Fold the two uncovered sections into the center, overlapping them slightly where they meet. You’ll now have a rectangle that’s about 9” x 12”.

Lightly score the dough into three 9” x 4” sections. Spread the remaining sauce over the right two sections and top with the remaining cheese. Fold the left (un-sauced) section over the center section and then fold the right (sauce-covered) section over both, like you'd fold a letter.

Transfer the loaf seam-side up to the prepared pan and pierce the top with a fork about a dozen times.

Bake the loaf for 50 to 60 minutes, until the top is dark and blistered in places; a digital thermometer inserted into the center of the loaf will read 212°F. Remove the loaf from the oven and turn it out of the pan onto a rack.

Allow the scaccia to cool slightly (for about 10 minutes) before cutting it into thick slices. Serve warm or at room temperature.

Store leftover scaccia, well wrapped, for 2 to 3 days at room temperature; freeze for longer storage. Toast slices or reheat to serve.

Pour the tomatoes and their juice into a blender and purée until smooth. In a small saucepan, warm the tablespoon olive oil over medium heat. Add the garlic and cook, stirring, until fragrant and barely beginning to brown, about 2 minutes. Pour the tomatoes into the saucepan along with the teaspoon of sugar, bring to a simmer, and cook, stirring occasionally, until the sauce thickens slightly, about 10 minutes. Remove the sauce from the heat, stir in the basil, and season liberally with salt and pepper.