Remove tough stems from mushrooms and tear large ones into quarters, smaller ones in half
Bring a large pot of salted water to boil for pasta
Preheat a large pot over high heat for about 3 minutes until very hot
Add 3 tablespoons olive oil and thyme to the hot pot until smoking
Add all mushrooms at once and let cook without stirring for about 5 minutes until water evaporates
Stir mushrooms and continue cooking for about 10 minutes total until golden brown
Drop pasta into boiling water and set timer for 2 minutes shy of al dente
Add chopped shallots and garlic to mushrooms, reduce heat to medium-low, and stir constantly for a few minutes until softened
Turn off heat under mushroom mixture
Reserve 2 cups of pasta cooking water before draining pasta
Add drained pasta to mushroom pot with about 1 cup of pasta water
Return pot to medium heat and cook until water is mostly evaporated, about 3 minutes
Turn off heat and let cool for about 3 minutes
Whisk together egg yolks, whole egg, grated Parmesan, and black pepper
Temper eggs by slowly adding ½ cup hot pasta water to egg mixture while whisking
Slowly stream tempered egg mixture into pasta pot while stirring in circles to prevent scrambling
Add reserved pasta water as needed to reach desired sauce consistency
Stir in chopped fresh parsley
Taste and adjust seasoning as needed
Serve immediately
