Heat a skillet over medium heat and add the butter. Once melted, add the sliced mushrooms and cook for about 4-5 minutes, stirring occasionally, until golden and tender.
Add the minced garlic and fresh thyme to the mushrooms. Cook for an additional minute. Season with a little salt to taste.
Remove the mushroom mixture from the skillet and stir in the chopped spinach. The residual heat from the mushrooms will help soften the spinach. Set the filling aside.
Add another small pat of butter or a drizzle of oil to the same skillet over medium heat. Whisk together the eggs and milk, then pour into the skillet. Gently spread the eggs out into a thin, even layer about the size of your tortilla.
While the eggs are still slightly wet on top, lay the tortilla directly over the eggs. Press down gently.
Once the eggs are fully set and the tortilla has bonded to them, carefully flip the whole thing over so the tortilla side is now down on the skillet. Season the egg side with salt and pepper to taste.
Add the shredded Gouda over the egg side. Let it melt for about 30-60 seconds.
Spoon the mushroom and spinach filling down the center of the wrap. Slide onto a plate, fold or roll it up, and enjoy immediately.