Heat the oven to 400 degrees.
Heat oil in large nonstick skillet. Add mushrooms, tomatoes and thyme. Cover the skillet, and cook over medium heat until the mushrooms are stewing well in their juices, 7 to 8 minutes. Uncover and cook, stirring frequently, until the mixture is nearly dry, about 2 to 3 minutes longer.
Raise the heat to medium-high, and, when the mixture is sizzling away, add the beaten eggs mixed with the salt. Stir every few seconds until the eggs are barely set and still very creamy. Remove immediately from the heat and stir in the optional goat cheese or cream cheese.
Wrap the tortillas in a damp paper towel and then in plastic wrap. Microwave on high until hot, about 45 seconds. Let stand covered for a few minutes.
Spoon ½ cup of the salsa over the bottom of a 11x7-inch baking dish. One by one, remove a hot tortilla from the cloth, spoon on a portion of the egg filling, roll up and place seam-side down in the baking dish. You’ll need 8 filled tortillas; there are two extra, in case some break up.
Spoon the remaining enchilada sauce over the enchiladas, making sure to completely cover the tortillas, then sprinkle with the cheddar cheese. Bake just until the cheese is melted and the sauce hot, about 20 minutes.
Let stand 5 minutes, then sprinkle with cilantro and serve.