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Heat 3 tablespoons oil in a large straight-sided skillet over medium-high. Add bread; cook, flipping once and adding sage sprigs halfway through, until toasted, 10 minutes. Transfer to a plate; lightly rub one side with whole garlic clove. Season with salt.
Discard oil; return skillet to medium-high heat. Add remaining 1 tablespoon oil, sliced garlic, carrots, and chopped sage; season with salt and pepper flakes. Cook, stirring, until garlic is golden. Add wine; cook until evaporated, 2 minutes. Stir in beans and broth. Bring to a boil, then reduce heat and simmer until carrots are tender and liquid has reduced slightly, 25 minutes. Transfer 1 cup beans to a bowl; mash and return to pot. Stir in greens just until wilted; season with salt. Serve over toast, with cheese and a drizzle of oil.
¼ cup extra-virgin olive oil, plus more for serving
4 slices (each 1 inch thick) ciabatta (from a 12-ounce loaf)
3 sprigs sage, plus 1 ½ teaspoons chopped fresh sage leaves
3 cloves garlic, peeled, 2 thinly sliced
Kosher salt and red-pepper flakes
3 medium carrots, peeled and sliced into ¼-inch rounds (¾ cup)
½ cup dry white wine, such as Sauvignon Blanc
2 cans (each 15 ounces) cannellini beans, rinsed and drained
4 cups low-sodium vegetable broth
4 cups assorted baby greens, such as kale, spinach, and chard
Grated Parmesan or Grana Padano, for serving