In a frying pan, melt 1 Tbsp of the butter over medium heat. Add the mushrooms and cook, stirring occasionally, until the juices evaporate and the mushrooms begin to brown, about 6 minutes. Stir in ½ tsp of the thyme and season with salt and pepper. Transfer to a bowl.
Preheat the oven to 200°F (95°C). In a bowl, whisk together the eggs, the cream, ¼ tsp salt, and a few grinds of pepper just until blended. Do not overbeat.
Add half of the remaining 1 Tbsp butter to the pan and melt over medium heat. Tilt the pan to cover the bottom evenly with butter. Add half of the egg mixture and cook until the eggs have barely begun to set around the edges, about 30 seconds. Using a heatproof spatula, lift the cooked edges and gently push them toward the center, tilting the pan to allow the liquid egg on top to flow underneath, then cook for 30 seconds. Repeat 2 more times. When the eggs are almost completely set but still slightly moist on top, sprinkle half of the cheese over half of the omelet. Scatter half of the mushrooms over the cheese.
Using the spatula, fold the untopped half of the omelet over the filled half to create a half-moon shape. Cook the omelet for 30 seconds, then slide it onto a heatproof serving plate. Keep warm in the oven. Repeat to make a second omelet. Sprinkle the omelets with the remaining ¼ tsp thyme and serve.