Using a slotted spoon, transfer the carrots to a plate. Add remaining the remaining 2 tablespoons unsalted butter, 1/3 cup packed brown sugar, 1/2 teaspoon ground ginger, 1/4 teaspoon ground cinnamon, and remaining 1/2 teaspoon kosher salt to the pan. Stir to combine, increase the heat to high, and bring to a boil. Cook until thick and syrupy, 2 to 3 minutes.