Whisk together the soy sauce, hoisin sauce, chili-garlic sauce, ginger, vinegar, and sesame oil in a large bowl. Remove ¼ cup to a medium bowl and whisk in the peanut butter and ¼ cup hot water. Cover and set aside for serving.
Put the steak in a shallow dish and pour the remaining marinade on top, flipping to coat. Cover and refrigerate for at least 30 minutes and up to 2 hours.
Brush a grill or grill pan with the vegetable oil and heat over medium-high heat. Remove the steak from the marinade and pat dry with paper towels. Grill 4 to 6 minutes per side for medium-rare (a thermometer inserted into the center should read 135˚F).
Transfer the steak to a cutting board and let rest 5 minutes. Meanwhile, cook the rice noodles according to the package directions. Slice the steak against the grain. To serve, fill the lettuce leaves with the noodles, steak, carrots, cucumber, and mint and drizzle with some of the peanut sauce. Serve any remaining peanut sauce on the side.