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.