A thick layer of mashed avocado acts as a creamy spread for the burger buns and a good balance to the triple-whammy topping: lime-marinated red onion, pickled jalapenos, and cilantro.
Heat oil in large nonstick skillet over medium heat. Cook onion with 1/4 teaspoon salt, covered, stirring occasionally, until tender, about 7 minutes. Transfer to a bowl, and toss with 2 tablespoons lime juice.
Heat grill to medium-high. Divide turkey into 4 equal pieces. Shape each into a 3 1/2-inch patty (about 3/4 inch thick). Brush both sides of burgers with oil, and season with 3/4 teaspoon salt. Grill until cooked through and an instant-read thermometer inserted into centers reaches 155 degrees, about 4 1/2 minutes per side (3 1/2 minutes if using a grill pan). Grill buns, cut sides down.
Meanwhile, mash avocado with a fork, and mix with remaining tablespoon lime juice and 1/4 teaspoon salt. Spread onto buns. Sandwich romaine, burgers, onion, jalapenos, and cilantro between buns.