Bring 1 cup water to a boil. Add dried mushrooms, cover and let stand about 20 minutes. Squeeze dry; strain liquid (a coffee filter is handy for this) and reserve.
In a food processor, mince garlic and rehydrated mushrooms. Add fresh mushrooms and thyme; continue to process until all is minced.
Heat oil in a 12-inch skillet over medium-high heat. Add mushroom mixture; saute until nearly all moisture has evaporated, 5 to 7 minutes. Turn off heat, stir in cheese, and season with salt and pepper. Set aside.
Mix honey and vinegar in a small bowl.
Adjust oven rack to lowest position and heat oven to 425 degrees.
Set breasts, skin side down, on a large, heavy, lipped cookie sheet. Brush with half of the honey-vinegar; generously salt and pepper. Turn breasts over. Push fingers under skin to make a pocket; stuff with mushroom mixture. Again brush with honey-vinegar and season with salt and pepper. Being careful not to crowd, arrange breasts on cookie sheet so thickest ends point outward. (Can now be covered and refrigerated overnight; return to room temperature before roasting.)
Roast until golden brown, adding water if necessary to keep pan drippings from burning, until a meat thermometer registers 160 degrees in the thickest portion of the largest piece, 30 to 45 minutes. Transfer chicken to a platter.
Scrape pan juices into a medium saucepan. Add reserved mushroom-soaking liquid and enough water to equal 1 1/2 cups of liquid. Bring to a simmer. Add cornstarch mixture; continue to simmer until it thickens a bit.
Arrange a breast on each of 8 plates; drizzle with sauce, and serve.
