In a medium bowl, whisk the eggs, cream, and thyme. Season with salt and pepper, then stir in the cheese.
Add the remaining 1/2 Tbsp of butter to the pan. Heat over medium-low heat until melted and hot.
Pour in the egg mixture. Cook until almost set at the top (I like to cover my pan with a lid to speed it up, but it's not necessary).
When the omelette is done, place the mushroom filling on one half of the omelet, fold the other half over.
Transfer to a plate and enjoy!