1. In a blender or food processor, whirl parmesan cheese, pine nuts, dried tomatoes, tomato paste, garlic, and 2 tablespoons reserved oil from dried tomatoes until mixture is a chunky paste, scraping down sides of bowl as needed.
2. In a large bowl, mix ground beef with beer, 1 teaspoon salt, and 1 teaspoon pepper until well combined. Divide meat mixture into 12 equal portions and flatten each portion into a 4-inch round.
3. Top each of 6 of the rounds with about 1 tablespoon of the dried-tomato pesto. Top with remaining 6 rounds, pressing edges together to seal.
4. Lay burgers on a barbecue grill over a solid bed of hot coals or high heat on a gas grill (you can hold your hand at grill level only 2 to 3 seconds); close lid on gas grill. Cook burgers, turning once to brown both sides, until meat around filling is no longer pink (cut to test; note that pesto is reddish brown), 6 to 8 minutes total. Also lay bun halves, cut side down, on grill and toast 1 to 2 minutes.
5. With a wide spatula, transfer burgers to bun bottoms. Add salt and pepper to taste. Top each burger with about 2 tablespoons feta cheese, if desired. Cover with bun tops. Serve with jalapeño mayonnaise to add to taste.