1. Make the flatbread dough: In a large bowl, dissolve yeast in 1½ cups lukewarm water. Add flours and mix with fingertips until a loose dough forms. Cover with plastic wrap and let rest for 20 minutes.
2. Sprinkle salt over dough, add crème fraîche, and knead until well mixed and dough pulls away from sides of bowl. (It will still be soft and wet.) Cover again and let rise for 30 minutes. Knead dough again to deflate. Cover and chill for 30 minutes.
3. Make the whipped tomato chèvre: Using a food processor or mortar and pestle, grind tomatoes into a pulp, then pour through a fine mesh strainer. Using an electric mixer or stand mixer fitted with whisk attachment, whip chèvre and cream. Slowly stream in tomato puree and mix until combined. Season to taste, then chill until ready to serve.
4. Assemble the flatbreads: Using a vegetable peeler or mandolin, shave squash into thin slices. Sprinkle with salt and set aside for 30 minutes. Divide dough into eight portions. On a floured work surface, use your fingers to press dough out into ¼-inch-thick ovals. Brush with oil and grill until lightly marked, about 3 minutes per side. Spread with tomato chèvre and top with squash and radish.