In nonstick skillet over medium-high heat, heat 1 tablespoon olive oil. Introduce button and shiitake mushrooms and cook, stirring constantly 2 minutes. Stir in 1/4 cup sherry and continue to cook until mushrooms are soft and liquid has reduced by half. Remove mushrooms with slotted spoon; set aside. Reserve mushroom sherry liquid; set aside.
In same skillet, still over medium-high heat, heat 1 tablespoon olive oil. Place chicken breasts and garlic in pan and brown chicken on both sides. Sprinkle with the thyme, pour in the chicken broth, lower the heat and cover. Simmer until chicken is cooked through and no longer pink, 10 minutes. Remove chicken from broth and let cool. Reserve broth; set aside.
Shred cooled chicken with two forks or by pulsing in a food processor. In large bowl, stir together shredded chicken, ricotta, 1/4 cup Gruyere, and mushroom mixture until well combined.
Combine reserved chicken broth and reserved sherry-mushroom liquid to equal 1/2 cup. Skim off any oil that rises to the top.
Bring a large pot of lightly salted water to a boil. Add pasta and cook for 8 to 10 minutes or until al dente; drain.
Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13 baking dish. In a small saucepan, melt butter over medium heat. Dump in flour all at once and stir quickly with whisk to make a roux. Add 1/4 cup sherry and reserved broth/mushroom liquid a little at a time, whisking constantly until smooth. Sprinkle in salt and pepper, reduce heat to low, and stir constantly until thickened. Slowly whisk in milk and 1/4 cup Gruyere until completely incorporated and smooth.
Stuff each pasta shell with about 2 tablespoons chicken mixture and place shells closely packed in a single layer in prepared baking dish. Spoon the sauce over the shells and sprinkle with Parmesan. Bake 30 minutes, or until heated through and bubbly. Serve immediately.