Preheat the oven to 350 degrees F. Meanwhile, pickle the carrot: In a pint-size jar, combine carrot, vinegar, sugar, and 1 teaspoon of salt. Cover and shake to dissolve the sugar and salt.
Place salmon filets on a foil-lined baking sheet. Drizzle with 1 tablespoon of olive oil. Place in oven and bake until just cooked through and flaky, about ten to twelve minutes.
Cut lime into quarters, juice one and set the rest aside. Whisk together lime juice, mayonnaise, and chili paste to make a dressing.
Arrange the bibb lettuce leaves on a plate. Divide the salmon, carrots, bean sprouts, and scallions evenly between each lettuce cup. Top with mayo dressing, cilantro, lime wedges, and hot sauce, as desired.