Whisk together mayonnaise, mustard, lemon zest and juice, and oil in a bowl; season with salt and pepper. Reserve 1/2 cup mixture; add chicken to remaining marinade and turn to coat. Let marinate at room temperature 20 minutes.
Preheat grill to medium and brush grates with oil. Remove chicken from marinade and grill, flipping occasionally, until cooked through, 10 to 12 minutes. Transfer to a cutting board. Let stand 5 minutes; slice. Meanwhile, grill salami until lightly marked, 1 to 2 minutes a side.
Divide frisee among plates. Top with chicken, salami, avocado, and cucumber. Season with salt and pepper, drizzle with reserved marinade, and serve immediately.