Preheat oven to 500°F. Place pizza stone on lowest rack in oven during preheating.
Combine 1 cup all-purpose flour, cake flour, undissolved yeast, sugar and salt in a bowl. Add water and mix until well blended, about 1 minute. Add additional flour, if needed, to form a soft dough.
Knead** on a floured surface, adding additional flour if necessary, until smooth and elastic, about 4 minutes. Divide dough into 8-equal size portions. Cover dough while working with individual portions.
Heat oil and butter in large skillet over medium-high heat. Add mushrooms and radicchio; sauté 7 to 10 minutes or until mushrooms are softened and radicchio caramelized. Salt and pepper to taste. Set aside.
Roll one portion of dough on floured surface into a 12 x 4-inch rectangle. The dough should be extremely thin at this point.
Place flat bread on a well-floured or cornmeal-dusted pizza peel, flat baking sheet or the back side of a jelly roll pan. Parchment paper (not waxed paper) also works. Make sure that the flat bread moves freely. Add 1/4th of the radicchio over the flat bread. Sprinkle with 1 torn sage leaf, 1/4 cup fontina and 2 tablespoons Parmesan cheese. Repeat on additional three flat breads; you will have 4 flat breads left over.
To bake, slide the flat bread off the pizza peel or baking sheet onto the preheated stone that is still in the oven. Use a long-handled spatula to help transfer the flat bread if needed. Bake 3 to 5 minutes or until flat bread is lightly browned.
When the flat bread is done, remove it from the oven with a pizza peel. You can also use a spatula or tongs to gently transfer the flat bread to a cutting board, pizza pan or cookie sheet for serving.
Repeat with remaining dough portions. Any unused dough can be stored in the freezer until later use; wrap each portion individually in plastic wrap. Store up to 2 weeks in freezer. Thaw at room temperature about 20 minutes or in refrigerator about 1 hour.
Allow the oven to cool before removing the pizza stone and follow the manufacturer’s directions for cleaning.
*If you don't have a thermometer, water should feel very warm to the touch.
**To knead the dough, add just enough flour to the dough and your hands to keep the dough from sticking. Flatten dough and fold it toward you. Using the heels of your hands, push the dough away with a rolling motion. Rotate dough a quarter turn and repeat the "fold, push and turn" steps. Keep kneading the dough until it is smooth and elastic. Use a little more flour if dough becomes too sticky, always working the flour into the ball of the dough.
NOTE: While a pizza stone is preferred for making flat bread, you can also use a heavy baking sheet. Preheat the baking sheet in a 500°F oven for 10 minutes. Transfer rolled flat bread to hot baking sheet using the directions above for transferring the dough to a pizza stone.
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