To make the starter: Mix the 1 ¼ 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.
To make the dough: Combine the risen starter with the water, salt, flour, and yeast. Knead -- by hand, mixer to make a smooth dough -- about 8 minutes.
Place the dough in a lightly greased bowl, cover, and let it rise for 1 ½ to 2 hours, till it's nearly doubled in bulk.
Gently deflate the dough, and pat and stretch it into a ¾"- thick rectangle, about 9"x 12". Spritz with water, and sprinkle with the grated cheese.
Starting with a long side, roll it into a log, pinching the seam to sea. Place the log, seam-side down, on a lightly floured or lightly oiled surface.
Cover it and let it rise for 1 to 1 ½ hours, till it's puffy though not doubled in bulk. Towards the end of the rising time, preheat the oven to 425 F.
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 a lightly greased or parchment-lined baking sheet, 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.
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 deep golden brown. Remove them from the oven, and cool on rack.