Preheat oven to 425°F. Mist 6 shallow (approximately 12-oz.) baking dishes with cooking spray. Place on a large baking sheet.
Warm oil in a large skillet over medium-high heat. Add onion and peppers; cook, stirring often, until softened but not browned, about 10 minutes. Add garlic and sauté 1 minute longer. Stir in tomato paste and 1 Tbsp. water. Taste and season with salt and pepper.
Divide vegetable mixture among baking dishes. Crack two eggs into each dish; sprinkle with cheese and season with salt and pepper. Bake until egg whites are set but yolks are still runny, about 15 minutes. Let stand 5 minutes before serving to allow yolks to firm up a little.