Preheat the oven to 375 degrees F. Wrap each roll in foil and bake until crisp on the outside, about 15 minutes.
Meanwhile, toss the carrot, cucumber, vinegar and 2 teaspoons fish sauce in a bowl; set aside, stirring occasionally, 10 minutes. Combine the mayonnaise, Sriracha and the remaining 1 teaspoon fish sauce in a small bowl. Beat the eggs in a medium bowl.
Unwrap the rolls; spread the bottom buns with the mayonnaise mixture and top with the pate.
Heat the vegetable oil in a large nonstick skillet over medium heat. Add the eggs and cook, stirring, until just set, about 3 minutes; divide among the rolls. Add the basil and carrot-cucumber salad to the sandwiches and drizzle with some of the vinegar mixture from the salad bowl. Serve with potato chips; sprinkle the chips with vinegar and drizzle with Sriracha.