Opposite. You DON'T speed up the process, or the bottom will brown and be yucky. (There's no excuse for brown eggs! None! - I always was a rapier and hat with a feather when I say that. Avast! No brown blended eggs!!!)
For a giant omelette... you keep the temp very low. 3 or 4 max, 2 or 3 ideal. Only use veggie oil in the pan (no butter, but if you like butter, you can drizzle some over once it's done cooking for the flavor). Keep a lid covering it. Once the bottom has begun to solidify (in about 5-10 minutes) use your spatula to scoot back the cooked egg, so raw egg can run under. Do this on all sides. Replace the lid. Let it go until the top is glossy. Run your spatula down the middle (not to cut, but to indent) in 2 parallel lines about an inch apart (for ease in folding). Place most of your cooked filling on half of the omelette, and smother with, or wave cheese in the general direction of them. Slide it off the pan onto a platter, flipping the 'bare' half over the filling. Sprinkle remaining toppings over the top. Drizzle with herb oil or butter. Finis.
((While the omelette is cooking, have the filling cooking in another pan beside it, so it's hot and ready to go when your omelette is done cooking)).
For a VERY fluffy omelette:
Separate out half of your egg whites. Mix the yolks & half the whites together. Then whip the other half of your whites into soft peaks (stiff peaks are too heavy duty to mix well). Fold the egg whites gently into your yolk and white omelette.