1) To make the starter: Mix the 1¼ cups flour, salt, yeast, and ½ cup water in a medium-sized bowl. Mix till well combined; the starter will be very dry. Cover and let rest overnight at room temperature; it'll become bubbly.
2) To make the dough: Combine the risen starter with the water, salt, flour, and yeast. Knead—by hand, mixer, or bread machine set on the dough cycle—to make a smooth dough.
3) Place the dough in a lightly greased bowl, cover, and let it rise for 1½ to 2 hours, till it's nearly doubled in bulk.
4) Gently deflate the dough, and pat and stretch it into a 3/4"-thick rectangle, about 9" x 12". Spritz with water, and sprinkle with the grated cheese.
5) Starting with a long side, roll it into a log, pinching the seam to seal. Place the log, seam-side down, on a lightly floured or lightly oiled surface.
6) Cover it and let it rise for 1 to 1½ hours, till it's puffy though not doubled in bulk. Towards the end of the rising time, preheat the oven to 425° F.
7) Gently cut the log into four crosswise slices, for mini-breads; or simply cut the dough in half, for two normal-sized loaves. Place them on one (for two loaves) or two (for four mini-loaves) lightly greased or parchment-lined baking sheets, cut side up. Spread them open a bit, if necessary, to more fully expose the cheese. Spritz with warm water, and immediately place them in the preheated oven.
8) Bake for 20 minutes (for the mini-loaves), or 35 minutes (for the full-sized loaves), or until the cheese is melted and the loaves are a very deep golden brown. Remove them from the oven, and cool on a rack.