In a small bowl, stir together gochujang, rice vinegar, honey, and 1 tablespoon water until smooth. Set aside. This keeps for weeks in the fridge if you want to double it.

Heat a large skillet over medium-high heat. Add the ground beef and cook, breaking it up, until no pink remains and some edges are crispy, about 8 minutes. Drain excess fat if needed, then stir in garlic, soy sauce, brown sugar, and sesame oil. Cook 1 minute more until glossy. Transfer to a bowl and wipe out the skillet.

Return skillet to medium heat with 1 teaspoon oil. Add spinach by the handful, stirring until just wilted, about 2 minutes. Season lightly with salt, squeeze out excess moisture, and transfer to a plate.

Add remaining 1 teaspoon oil to the skillet. Add carrots and cook over medium-high heat until just tender with a little char, about 3 minutes. Season with a pinch of salt and transfer to the plate with spinach.

Add a little more oil to the skillet if needed. Fry eggs sunny-side up or over easy - you want runny yolks to mix into the rice. Season with salt and pepper.

Divide warm rice among 4 bowls. Arrange beef, spinach, and carrots in separate piles. Top each with a fried egg, drizzle generously with sauce, and sprinkle with sesame seeds. Serve immediately with extra sauce on the side.