How to Make It
Preheat oven to 325°.
Cut off top 1/4 inch of garlic heads to expose cloves. Place garlic, cut side up, on a piece of foil. Drizzle with 1 tablespoon oil. Wrap garlic heads tightly in foil. Roast at 325° for 1 hour and 15 minutes or until very soft; cool slightly. Squeeze pulp from roasted garlic into a small saucepan; discard papery skins. Add 3/4 cup half-and-half to pan. Bring just to a boil, stirring to mash garlic; remove from heat. Add potato flakes, stirring with a whisk.
Arrange potatoes on a foil-lined rimmed baking sheet coated with cooking spray; coat potatoes with cooking spray. Roast at 325° for 45 minutes or until thoroughly cooked, stirring every 15 minutes; set aside.
Heat a large skillet over medium-high heat. Add 1 tablespoon oil to pan; swirl to coat. Add bell pepper, carrot, thyme, and 1/4 teaspoon salt to pan; cook 2 minutes, stirring frequently. Reduce heat to medium-low; cook 12 minutes or until soft. Remove mixture from pan; set aside.
Wipe pan with paper towels; return to medium-high heat. Add 1 tablespoon oil to pan; swirl to coat. Add onion and 1/4 teaspoon salt. Cook 3 minutes, stirring frequently. Reduce heat to medium-low. Cook 15 minutes or until caramelized and soft, stirring occasionally. Set aside.
Heat a large cast-iron skillet over medium-high heat. Add remaining 3 tablespoons oil to pan. Add potatoes; cook 4 minutes or until golden brown. Stir in bell pepper mixture, onion, steak, and 3/4 teaspoon salt. Cook 1 minute or until thoroughly heated. Stir in garlic cream mixture, 2 tablespoons parsley, and 1/4 teaspoon pepper; remove pan from heat.
(Optional) Add water to a saucepan, filling two-thirds full; bring to a boil. Reduce heat; stir in vinegar. Break each egg into a custard cup. Gently pour eggs into pan; cook 3 minutes. Remove with a slotted spoon; place on a paper towel-lined plate.
Stir remaining 2 tablespoons half-and-half into hash to loosen. Divide hash among 6 plates. Top each serving with 1 egg, if desired; sprinkle evenly with remaining 1/4 teaspoon salt and 1/4 teaspoon pepper. Garnish with remaining 2 tablespoons parsley.