For the mayonnaise: In a small bowl, mix together mayonnaise, garlic, and rosemary; set aside.
For the burgers: Preheat a gas or charcoal grill or place a grill pan over medium-high heat. In a large bowl, add the ground chicken, 1/2 teaspoon salt, 1/4 teaspoon pepper, and 1/2 of the mayonnaise mixture. Using clean hands, gently combine the ingredients and form the chicken mixture into 4 patties. Place the burgers on the grill and cook for about 7 minutes on each side. Transfer to paper towels and let rest for a few minutes.
Brush the cut side of each roll with the olive oil and 1 teaspoon of the mayonnaise mixture. Grill for 1 to 2 minutes until slightly golden.
To assemble the burgers: Spread a dollop of the remaining mayonnaise mixture on the tops and bottoms of the toasted buns. Place the chicken burgers on the bottom halves of the buns. Top each with 1/4 cup of arugula and finish with the top half of the bun.