If you have extra mayoinnaise, the herby, lemony spread is great on sandwiches and burgers (or you could just make more hot dogs).
Process mayonnaise, basil, lemon zest, and garlic clove in a blender or food processor until smooth, pausing as needed to scrape down sides. Toss together avocado, cherry tomatoes, salt, and pepper in a small bowl; chill until ready to use.
Preheat grill to medium-high (350°F to 400°F). Grill hot dogs about 6 minutes, turning occasionally, until heated through and grill marks appear. Grill buns, if desired.
Spread cut sides of buns with about 1 tablespoon of the mayonnaise mixture. Place hot dogs in buns; top evenly with avocado mixture.