Stir cucumber pieces with 1/4 cup gochujang paste in a bowl; set aside.
Bring about 2 cups of water to a boil in a large nonstick skillet and stir in spinach; cook until spinach is bright green and wilted, 2 to 3 minutes. Drain spinach and squeeze out as much moisture as possible; set spinach aside in a bowl and stir soy sauce into spinach.
Heat 1 teaspoon olive oil in large nonstick skillet and cook and stir carrots until softened, about 3 minutes; stir in garlic and cook just until fragrant, about 1 more minute. Stir in cucumber pieces with gochujang paste; sprinkle with red pepper flakes, and set the mixture aside in a bowl.
Brown beef in a clean nonstick skillet over medium heat, about 5 minutes per side, and set aside. In a separate nonstick skillet, heat 1 more teaspoon olive oil over medium-low heat and fry the eggs just on one side until yolks are runny but whites are firm, 2 to 4 minutes each.
To assemble the dish, divide cooked rice into 4 large serving bowls; top with spinach mixture, a few pieces of beef, and cucumber mixture. Place 1 egg atop each serving. Drizzle each bowl with 1 teaspoon of sesame oil, a sprinkle of sesame seeds, and a small amount of gochujang paste if desired.