On a work surface, pound the thighs 1/4 inch thick.
Melt 2 tbsp. butter in a large skillet over medium heat. Add the breadcrumbs and cook, stirring, until toasted, about 5 minutes; transfer to a bowl. Add another tbsp. butter, the shallot, celery and thyme to the skillet; cook 2 minutes. Add the apple; cook until tender, 2 minutes. Add the cranberries and 1/2 cup broth; season with salt and pepper. Transfer the mixture to the same bowl, mix and wipe out the skillet.
Divide the stuffing among the thighs; roll up and tie with kitchen string. Melt the remaining 1 tbsp. butter in the skillet over medium heat; add the chicken and season. Cook, turning, until golden, 8 to 10 minutes. Pour the remaining 1/2 cup broth and 1 cup water into the pan, scrape up any brown bits and bring to a simmer. Add the mushrooms, partially cover and simmer, shaking the pan occasionally, until the mushrooms soften, about 15 minutes.
Transfer the thighs to a cutting board. Cook sauce over medium heat until slightly thickened, 1 to 2 minutes. Remove the string from the thighs. Slice each roll crosswise and transfer to plates. Drizzle with the pan sauce.