To prepare the mushrooms, slice the oyster mushrooms to 1/4-in. thick. Quarter or halve chanterelles lengthwise to make pieces of a similar size. Slice lobster mushrooms 1/8-in. thick.
Heat a large sauté pan over medium-high heat for 20 - 30 seconds. Glaze the bottom of the pan with oil.
Lay in the oyster mushrooms, taking care not to crowd the pan. Sprinkle the mushrooms with salt and add a few sprigs of thyme. Sauté, turning the mushrooms to brown them on both sides, about 6 minutes. Remove the mushrooms from the pan and set aside on a plate.
Check the pan to see if you need to add another glaze of oil. Repeat the sautéing of the mushrooms, cooking each variety independently, and seasoning each batch with salt and thyme.
When you're finished sautéing all of the mushrooms, add another glaze of oil to the pan if needed and add the shallots and a sprinkle of salt. Cook for 30 seconds, then add the garlic and cook for another 30 seconds.
Add all of the mushrooms and the butter and stir. When the butter has melted, remove from heat and allow to cool.
Preheat oven to 400°F/205°C.
Place boned chicken, skin-side down, on cutting board. Using a 3-in. paring knife, create pocket in breast meat and open. Cover with plastic wrap or parchment paper and using a mallet, flatten meat. Season meat with salt and pepper.
Place 2 - 3 Tbsp. of cooled roasted mushrooms down the length of the breast and cover them with flattened breast meat to seal pocket. Season chicken with salt and pepper. Repeat with remaining chicken breasts.
Heat a 5-qt. sauté pan over medium heat and add 2 Tbsp. olive oil.
Add breasts, flesh-side down and sear for 3 - 4 minutes. Turn over breasts and continue to sear on skin side. Optional: Weigh chicken down with the foil-covered bricks.
Transfer sauté pan to preheated oven and roast for 10 - 15 minutes or until thickest part of breast reaches 160°F/71°C.
Remove chicken from oven and let rest for 5 minutes before serving.