Step 1

Finely grate the zest of 1 medium lime (about 2 teaspoons). Juice the lime. Transfer half the zest and 2 tablespoons of the juice to a food processor fitted with the blade attachment. Transfer the remaining zest and 1/2 teaspoon of the juice to a small bowl.

Step 2

Cut 1 pound raw peeled and cleaned shrimp into 1-inch pieces. Trim and cut 2 medium scallions into 1-inch pieces. Transfer both to the food processor. Sprinkle with 1 1/2 teaspoons kosher salt and 1/2 teaspoon garlic powder. Pulse until coarsely chopped, about 8 (1-second) pulses, scraping down the sides of the bowl halfway through.

Step 3

Transfer to a large bowl. Add 1 large egg and sprinkle with 1/2 cup panko breadcrumbs. Stir and smash against the side of the bowl with a flexible spatula until combined. Halve, pit, and peel 1 ripe but firm medium California Avocado. Cut into small dice (about 1/4-inch), add to the bowl, and gently fold until just combined.

Step 4

Line a baking sheet with parchment paper or a silicone baking mat. Divide the mixture into 4 portions (about 6 ounces each). Gently shape each portion into a 3 1/2-inch wide patty about 3/4-inch thick and place in a single layer on the baking sheet. If any pieces of California Avocado fall out, just push them back in. Refrigerate while you prepare the toppings and buns, at least 15 minutes.

Step 5

Add 1/4 cup mayonnaise and 1 teaspoon Dijon mustard to the reserved bowl of lime zest and juice and stir to combine. Refrigerate until ready to use. Cut 1 medium beefsteak tomato crosswise into 4 rounds about 1/2-inch thick.

Step 6

Split 4 brioche or potato buns. Heat 1 teaspoon of the vegetable or olive oil in a large nonstick frying pan (at least 12 inches wide) over medium heat until shimmering. Tilt the pan to coat the bottom with oil. Place 4 bun halves cut-side down in the pan. Cook, moving the buns around occasionally so they cook evenly, until toasted and light golden brown, 2 to 3 minutes. Transfer to a plate. Repeat toasting the remaining bun halves in another 1 teaspoon oil. Transfer to the plate.

Step 7

Heat the remaining 2 tablespoons vegetable or olive oil in the same pan over medium heat until shimmering. Using a flat spatula, transfer the shrimp patties into the pan and cook until golden brown, cooked through, and the centers register at least 120ºF on an instant-read thermometer, 3 to 4 minutes per side. Spread the lime mayonnaise on the cut-sides of all the buns while the second side is cooking.

Step 8

Assemble the burgers: Place a shrimp patty on each bottom bun. Top each with a slice of tomato and 1 large butter lettuce leaf, and close with the top bun.