Heat 1 tablespoon oil in a large skillet over medium-high heat. Add leek; cook 2 minutes, stirring. Add mushrooms and thyme; cook 4 minutes.
Heat a cast-iron skillet over medium-high heat. Add remaining 3 tablespoons oil and potatoes; cook 4 minutes. Add leek mixture, garlic cream, sprouts, tomatoes, 1 teaspoon salt, and 1/4 teaspoon pepper; cook 2 minutes. Stir in half-and-half.
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; sprinkle evenly with remaining 1/4 teaspoon salt and 1/4 teaspoon pepper. Garnish with parsley.