1Preheat a grill to medium-high heat (400°F) or heat a grill pan over a medium heat until hot. Mix mustard and mayonnaise and set aside. Combine chicken, shallot, salt, and pepper. Form mixture into 4 patties. Rub burgers with oil. Press a deep dimple into the center of each burger (this will help it stay flat while it cooks).
2Grill burgers, turning once halfway through, until cooked all the way, about 10 minutes total. In last 3 minutes, top each burger with two slices of ham and a slice of cheese. Cover, and let cook remaining time until cheese is melted.
3Assemble burgers, spreading top bun with some of the reserved mustard mixture and a few arugula leaves.