How to Make It
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.