Preheat gas or charcoal grill for direct medium heat. Tear off four 18x 36-inch pieces of heavy-duty foil. Fold in half to make 18x18-inch squares. Brush with olive oil. Place a portobello mushroom cap, gill side up, in center of each piece of foil. Top with half of potato slices, and half of onion slices. Sprinkle with salt and black pepper. Place 1/2 tablespoon butter on each stack. Top with remaining potato and onion slices. Sprinkle with salt and black pepper; top each stack with another 1/2 tablespoon butter. Top with sprigs of fresh tarragon. Carefully bring foil sides up to cover vegetable stacks. If needed, insert large skewer through vegetables to secure.
Grill vegetable stacks over direct heat about 30 minutes or until vegetables are tender.
Meanwhile, brush both sides of beef tenderloin filets with olive oil; season with salt and black pepper. Grill over direct heat 10 to 12 minutes for medium-rare doneness (145°F) or 12 to 15 minutes for medium doneness (160°F).
Transfer filets to serving platter. Immediately cut 1/4-inch slices from Whiskey Butter log; top filets with butter slices. Cover; allow to rest for 5 minutes. Carefully open foil packets and use spatula to transfer vegetable stacks to platter, restacking as needed. Sprinkle vegetables with 1 tablespoon snipped fresh tarragon. Just prior to serving, top filets with additional butter, if desired. Makes 4 servings.
*Make Ahead Tip: Compound butter can be made ahead, wrapped well, and refrigerated up to one week ahead or frozen for up to one month.