In a medium bowl, whisk the eggs, heavy cream, and a pinch of salt together until foamy.
Heat a small nonstick pan over medium heat, and add the butter. Once the butter begins to bubble, add the asparagus and season with both salt and pepper. Saute until softened. Set asparagus aside on a plate and tent with foil.
If your pan needs more butter, add another dollop. Add the eggs, and reduce the heat of your pan to low.
Allow the eggs to set for about 20 seconds, then use a rubber spatula to lift the eggs, and allow the liquid eggs to fill in the gaps. Repeat until your omelette is no longer runny, but still wet in the center.
Sprinkle half of the omelette with cheese, and your set aside asparagus. Place a lid on top. Allow the omelette to cook this way for about 30 seconds, you're ready to remove the lid once the eggs are more set (cheese may not melt yet).
Use 2 rubber spatulas to fold the omelette over itself, then slide the omelette onto a plate. Drizzle pesto over the omelette before serving, and enjoy!