Grease non-stick muffin tin with butter. Cut away sides of 12 muffin liners, place circle into each greased cup. Spray with cooking spray one more time.
In bowl, combine potatoes, onion, egg, flour and salt. Divide among muffin cups, pressing evenly up sides and over bottoms. Coat lightly with cooking spray. Bake in 400°F oven for 20 minutes or until crisp browned at edges.
In large skillet, melt butter. Sauté onions, carrots first, then add garlic, mushrooms, basil and salt until softened. Remove from heat and add cheddar, bread crumbs, feta and eggs. Divide among potato lined cups. Top each with a few thin mushroom slices and sprinkle with sesame seeds.
Return to oven, reduce heat to 350°F and bake for 25 minutes. Carefully remove paper bottom before serving.