Pour the milk and water into the bowl of the stand mixer. Add the yeast, flour, butter, ricotta, and salt. Stir just until the dough comes together.
Using the dough hook attachment, mix the dough on medium speed (4 on the KitchenAid) until it’s very supple, smooth, and elastic, about 10 to 12 minutes. (My old 300 watt mixer started dancing across the counter at that speed, so I cut it back to 2 and added 3 minutes to the kneading time.)
Transfer the kneaded dough to the prepared 2-quart container. Use a piece of masking tape to mark the point at which the dough will have doubled in volume. Put the lid on top and leave to rise at room temperature (70 to 75° F) until the dough doubles and inflates into a dome, about 1 to 1-½ hours. It should deflate slightly when pressed.
Turn the dough out onto a lightly floured counter. Use the bench scraper to cut the dough into two equal pieces. Roll each piece around the counter, shaping into a ball. Cup your hands around the ball and move in tight circles, until the dough surface becomes taught and smooth. Place the shaped loaves smooth side up on a sheet of parchment paper, about 4 inches apart. Cover with plastic wrap.
Allow the loaves to rise at room temperature (70 to 75° F) until the look puffy and have nearly doubled in size, about 1 to 1-½ hours. The dough should spring back slowly when you press a finger into it.
Preheat the oven: About 1 hour before baking, put the baking stone in the middle of the oven and the cast iron skillet on the bottom rack. Heat the oven to 400° F.
Put a fresh piece of parchment on the bakers peel/rimless baking sheet and dust with flour. Gently flip the rounds onto the parchment so that the bottom side is up. Slide the parchment with the loaves onto the baking stone. Pour the ice cubes into the hot skillet and close the oven. Bake until the rounds are golden, about 20 to 30 minutes.
Cool and Store
Slide the peel/rimless baking sheet under the parchment and remove the loaves from the oven. Slide onto a wire rack and allow to cool for a few minutes, then remove the parchment and allow the loaves to cool completely, about 1 hour. Store in a resealable plastic bag at room temperature. Will keep for 2 to 3 days, or freeze for up to 1 month.