Heat a tablespoon of olive oil in a large skillet over medium heat. Add the finely chopped onion and sauté for 4-5 minutes until softened.
Stir in the grated carrot and cook for another 2 minutes, allowing it to become tender.
Add the minced garlic to the skillet and cook for 1-2 minutes until fragrant.
Add the thinly sliced cabbage to the pan, stirring to combine. Season with salt and pepper. Cook the mixture for 8-10 minutes, stirring occasionally, until the cabbage becomes tender and slightly caramelized.
In a separate bowl, whisk the eggs until well combined. Pour the eggs into the skillet, stirring gently to incorporate them with the vegetables. Cook for another 3-5 minutes until the eggs are scrambled and fully cooked through.
Taste and adjust seasoning with additional salt and pepper if needed, then serve warm.