Heat 1 tablespoon olive oil in a medium pot. Add mushrooms, season with a little salt and pepper, and cook over medium high heat until mushrooms just start to soften.
Add the chicken broth and thyme, bring to a gentle simmer.
As the broth is heating:
In a large pan, sweat onions in remaining olive oil until translucent. Add garlic and rice, stir well.
Add half of the wine, stirring until wine is almost completely absorbed by the rice. Repeat.
Add simmering stock, about ½ cup at a time, stirring constantly. As rice absorbs one ladle of stock, add another and continue stirring.
Once rice is almost cooked, season with salt and pepper to taste.
Stir in grated cheeses, just until well incorporated and smooth. Remove from heat, allow to cool to room temperature.
Whisk egg, add to the cooled risotto mixture along.
Cover with plastic wrap and chill for several hours - ideally overnight.