Heat 1 tablespoon olive oil in a skillet over medium heat. Add garlic and cook, about 1 minute. Add spinach and steam, covered, until wilted, about 1 minute. Season with salt. Transfer to plate.
Rinse pan and fill with 2 inches water; bring to a boil. Add carrot and cook until tender, about 1 minute. Transfer to plate. Reduce heat to a simmer and poach eggs, 3 to 4 minutes.
Divide quinoa between bowls. Top with egg, spinach, carrot, and cucumber. Whisk vinegar and 2 tablespoons olive oil; season with salt. Drizzle over bowls. Sprinkle with red chili flakes and chives.