Generously season tenderloin with salt, pepper and rosemary. Grill the tenderloin over direct medium heat**, with the grill lid closed, about 15-20 minutes, turning every 5 minutes, until meat reaches an internal temperature of 150°F-155°F.
Meanwhile, melt 2 tablespoons butter in skillet over medium heat. Add shallots and mushrooms and sauté until tender, adding additional butter as needed. Add 1 cup blue cheese and remaining butter; stir until smooth and slightly thickened. Add cayenne pepper, to taste.
Remove tenderloin from grill and let rest 5 minutes, then slice. Pour sauce over slices and sprinkle with remaining blue cheese.
*Any variety of mushrooms or a mixture can be used (oyster, shitake, baby portabella, crimini, etc.).
**To roast the tenderloin, heat oven to 425°F. Place tenderloin on roasting pan in oven and roast 12-15 minutes, until meat reaches an internal temperature of 150°F-155°F. Remove from oven and let rest 5 minutes before slicing.