To make Whiskey Butter: In a small bowl let shallot stand in whiskey for 5 minutes. Drain and discard whiskey. In a medium bowl beat 1/2 cup butter with an electric mixer on medium speed until smooth and slightly fluffy. Add shallot and parsley; beat to combine. Stir in garlic, salt and white pepper. Mix well; adjust seasoning to taste. Place butter on plastic wrap or wax paper. Form butter into round log about 1 1/2-inches in diameter. Roll butter in plastic wrap or wax paper. Refrigerate butter until firm and easy to slice, about 2 1/2 hours.*
Preheat gas or charcoal grill for 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 the center of each piece of foil. Top with half of the potato slices, and half of the onion slices. Sprinkle with salt and black pepper and place a 1/2 tablespoon butter on each stack. Top with remaining half of the potato slices and onion slices. Sprinkle with salt and pepper, and top with the remaining butter pieces. Top with sprig of fresh tarragon. Carefully bring foil sides up to cover vegetable stacks. If necessary, insert a 6-inch skewer through the 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 or 12 to 15 minutes for medium doneness.
Transfer filets to serving platter and immediately cut 1/4-inch slices from whiskey butter log; top filets with butter slices. Cover, allow to rest 5 minutes. Carefully open packets and transfer stacks to platter with spatula, restacking as needed. Sprinkle vegetables with 1 tablespoon snipped fresh tarragon.
Well-wrapped compound butter can be made ahead and refrigerated up to one week ahead or frozen for up to one month.