1. Preheat the oven to 375 degrees F. Line a rimmed baking sheet with nonstick aluminum foil.
2. Grate the carrot on the small holes of a box grater into a large bowl. Add the ground turkey, breadcrumbs, parsley, paprika, eggs, scallions, garlic, 1 teaspoon salt and 1/2 teaspoon pepper. Mix with your hands until just combined.
3. Divide the mixture into 6 portions and form each into a small loaf shape on the nonstick foil. Mix the steak sauce and brown sugar in a small bowl and brush over the loaves. Bake until just cooked through, 20 to 25 minutes. Serve warm.