Melt 1 tablespoon butter in a small skillet over medium-low heat. Add green onions and jalapeño and cook until softened, about 3–4 minutes. Remove from heat and allow to cool.
Finely dice half the shrimp into ¼-inch pieces. Place remaining shrimp in bowl of a food processor fitted with a metal blade and process until smooth, about 10 seconds.
Combine puréed and chopped shrimp in a medium bowl. Add cooled scallions and jalapeños, Worcestershire sauce, lemon zest, 1 teaspoon salt, and ½ teaspoon pepper. Mix thoroughly with clean hands or rubber spatula.
Divide mixture into four 5-ounce patties about ¾-inch thick. Arrange on a baking sheet or plate lined with parchment paper; cover with plastic wrap. Refrigerate at least 15 minutes.
Meanwhile, whisk together tartar sauce ingredients in a small bowl; season with ½ teaspoon salt and ¼ teaspoon pepper. Refrigerate until ready to use.
Melt 1 tablespoon butter in a large nonstick or cast-iron skillet. Add buns cut-side down and toast until golden brown, 1–2 minutes; wipe out skillet.
Add remaining 1 tablespoon butter and olive oil to now-empty skillet; melt over medium heat. Once foam subsides, add shrimp patties, increase heat to medium-high and cook until golden brown and an instant-read thermometer inserted in centers register 145ºF (63ºC); about 5 minutes per side.
To assemble: spread 1 tablespoon tartar sauce over cut side of bottom bun, top with a piece of lettuce, shrimp patty, a slice of tomato, and a couple avocado slices. Spread 1 tablespoon tartar sauce on cut side of top bun and place on top.
Serve immediately with extra sauce on side.