Bring potatoes to a boil in a large saucepan filled with water. Reduce heat and simmer 10 minutes. Drain.
Heat oil in a large cast-iron skillet over medium-high heat. Add potatoes and green onions; cook 4 minutes (do not stir). Stir in beans and steak; cook 2 minutes. Remove pan from heat.
Heat a large nonstick skillet over medium heat. Crack eggs into pan; cook 1 minute. Cover pan; cook an additional 2 minutes or until desired degree of doneness.
Combine ketchup, 2 tablespoons water, vinegar, brown sugar, onion powder, salt, and pepper in a bowl, stirring until well blended. Combine remaining 1 tablespoon water and sour cream in a small bowl, stirring until smooth. Divide potato mixture among 4 bowls; drizzle 3 tablespoons ketchup mixture and 1 tablespoon sour cream mixture over each serving. Top with eggs and oregano.