In a bowl, whisk eggs, cream, and salt just until blended; do not overbeat.
Heat an 8-inch nonstick frying pan with sloping sides over medium-high heat for 1 minute. When hot, add butter and tilt pan to coat evenly. When butter is foamy, pour egg mixture into pan and swirl to spread it out to edges of pan.
As the egg mixture begins to set on the bottom, lift one edge with a heatproof flexible spatula and tilt the pan to let the uncooked mixture on top flow underneath. Continue lifting edges of omelet and tilting pan, working your way around all sides, until no more uncooked egg mixture will flow underneath and the top is just a little moist, about 2 minutes total.
Sprinkle cheese either in a strip down the middle, if you plan on folding the omelet into thirds, or over one half, if you're folding it in half. Loosen the omelet with the spatula. For a third-fold, fold one edge over strip of cheese in the middle; tip pan to slide omelet, folded edge first, onto a warm plate, flipping remaining edge over filling as omelet leaves pan. For a half-fold, tip pan to slide omelet, cheese side first, onto warm plate; flip bare half over cheese as omelet leaves pan.
Folding a plain cheese omelet:
As egg mixture sets on pan bottom, lift edge with a spatula and tilt pan to let uncooked egg flow underneath. Continue the process, working around pan sides, until no uncooked egg mixture flows underneath and top is still moist.
Sprinkle cheese in a strip down the center of the omelet, in line with the pan handle. With the spatula, fold one side over the strip of cheese.
Run spatula under omelet to loosen, and tip pan to slide omelet, folded edge first, onto a warm plate. Flip remaining edge over filling as omelet leaves pan.