FLATBREAD (PART 1): Pour warm water into a medium mixing bowl. With a wooden spoon, stir in yeast to dissolve. Let sit 10 minutes, until mixture starts to foam. Stir in salt, honey, and olive oil. Stir in flours to form a sticky dough. Turn out on to a lightly floured board and knead until smooth. Place in a large oiled bowl and cover with plastic or a kitchen towel. Let rise in a warm place until dough has doubled in size, about 1½ to 2 hours.
GRILLED CARROTS: Meanwhile, heat grill (charcoal, gas, or grill pan) on medium. Peel carrots and cut in half lengthwise. Drizzle with olive oil, salt, and pepper. Place carrots on grill and close lid (if using grill pan, cover with a lid or sheet pan). Grill, turning every 4 or 5 minutes, until softened and slightly charred. Transfer to a plate to cool. Cut on bias into 1-inch pieces.
BRAISED RADISHES: In a small sauté pan, combine radishes, butter, vinegar, and sugar. Cook over medium-high heat, until radishes are just tender. Uncover, raise heat to high, and glaze radishes until liquid reduces by half. Remove from heat and set aside to cool.
FLATBREAD (PART 2): When bread dough has doubled in size, gently remove from bowl using a rubber spatula or dough scraper. Place on an unfloured surface (the oil from the bowl should prevent dough from sticking).
Scatter marjoram and onion over the surface of the dough, and with the aid of the scraper or spatula, fold the dough in thirds, as if folding a letter. Give the dough a quarter turn and fold in thirds again. Flatten slightly and cut in half. Cut each half into four pieces.
With a floured rolling pin, roll each dough portion to a long, thin oval, roughly 4-by-9 inches in size. Brush or spatter oil on top of each rectangle. Gently lift, and place directly onto hot grill or grill pan. Grill over medium heat, until browned, 3 or 4 minutes. Flip with tongs or a spatula, and brown other side. Set aside to cool slightly.
SALAD: In a large bowl, combine grilled carrots, braised radish, and dandelion. Add mozzarella.
In a small bowl, whisk together olive oil and red wine vinegar, plus salt and pepper to taste. Pour vinaigrette over salad, and toss gently.
To serve, place one flatbread on each plate. Top with a handful of salad, making sure each has all elements, equally divided. Serve any extra flatbreads on the side or save for another use.