Fried Farro with Pickled Carrots and Runny Eggs

Place carrots and chile in a small heatproof bowl. Bring vinegar, sugar, 1 teaspoon salt, and 1/2 cup water to a boil in a small saucepan, stirring to dissolve sugar and salt. Pour over carrots and chiles; let sit at least 30 minutes. Drain, reserving 1/4 cup pickling liquid. Meanwhile, heat 1 tablespoon oil in a medium saucepan over medium-high heat. Add farro and half of garlic and cook, stirring constantly, until farro is dark brown, 8-10 minutes. Add 6 cups water and bring to a boil. Boil farro until tender but still firm to the bite, 25-30 minutes. Drain; let cool. Heat 1 tablespoon oil in a large skillet over medium-high heat. Add mushrooms and cook, tossing often, until soft and just starting to brown, 5-7 minutes. Using a slotted spoon, transfer to a plate. Add 2 tablespoons oil, then kale and remaining garlic to skillet. Cook, tossing often, until kale is wilted, about 4 minutes. Add colatura, farro, mushrooms, and pickled carrots. Cook, tossing often, until warmed through, 5-7 minutes; season with salt and pepper and reserved pickling liquid. Mix in parsley; divide among bowls. Heat remaining 1 tablespoon oil in a small nonstick skillet over medium-high heat. Crack eggs into skillet and fry until egg whites are set but yolks are still soft and runny, about 3 minutes (slightly longer for duck egg). Top farro with eggs and scallions; season with red pepper flakes. Just before serving, break up eggs and mix into farro. DO AHEAD: Carrots can be pickled 2 weeks ahead; cover and chill.