Heat oil in skillet over medium heat. Add green onion and garlic; cook, stirring frequently, until tender but not browned. Add carrots; cook until tender crisp, about 2 minutes more. Remove from heat.
Whisk eggs in large bowl. Add cheese, cornmeal, basil, nutmeg and pepper in large bowl; whisk to combine. Stir in carrot mixture.
Spray 12 (see note below) or 24 (1½-inch (4 cm) muffin cups with cooking spray. Spoon about 1 tbsp (15 mL) carrot mixture into each muffin cup. Bake in preheated 325°F (160°C) oven until set, about 15 minutes.
Let stand for 2 minutes. Remove quiches from pans. Serve hot.
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