Choose tall, robust dandelions and harvest in the morning before the flowers have fully opened. Grasp the yellow part of the flower (the petals) and twist away from the green sepals and stem.
Heat oil in a large skillet over medium-high heat. Add morels and cook, stirring, until tender, about 5 minutes. Remove from pan and set aside.
Season eggs with salt and pepper; pour mixture into pan. Stir gently with a wooden spoon to create curds. Pull cooked egg away from sides, allowing raw egg to flow underneath; cook until set, 4 to 5 minutes. Sprinkle Gruyere, dandelion petals, and morels over one half of the pan. Season with salt and pepper. Fold omelet in half. Garnish with a few petals.