Fried Farro with Pickled Carrots and Runny Eggs

Name: Fried Farro with Pickled Carrots and Runny Eggs
Ingredients: 2 medium carrots, peeled, chopped; ½ cup white wine vinegar; 1 tsp. sugar; 1 tsp. kosher salt plus more; 5 Tbsp. olive oil, divided; 1½ cups semi-pearled farro; 2 garlic cloves, chopped, divided; 6 oz. maitake mushrooms, torn into 1” pieces; ½ bunch Tuscan kale, center ribs and stems removed, torn into 1” pieces; 1 Tbsp. colatura (anchovy sauce) or fish sauce (nam pla or nuoc nam); Freshly ground black pepper; ½ cup fresh flat-leaf parsley leaves; 2 large eggs or 1 duck egg; 1 red chile (such as jalapeño or Fresno), seeded, coarsely chopped; 2 scallions, thinly sliced; Crushed red pepper flakes