Mix drained potato, 1 tsp. salt, 1/4 tsp. pepper, and chives in a medium bowl. Heat 1 tbsp. plus 2 tsp. oil in a large frying pan over high heat. Drop potato mixture into pan in 4 mounds and spread to form cakes. Cook until well browned on each side, turning once, about 5 minutes total.
Heat grill to medium (350° to 450°). Mix lamb with remaining 1/2 tsp. salt and 1/4 tsp. pepper, in a large bowl until just combined (do not overmix). Divide into 4 portions and form into 1/2-in. patties with a slight depression in the center of each to help them cook evenly. Brush with remaining 1 tsp. oil.
Grill burgers, covered, turning once, until done the way you like, about 7 minutes for medium-rare. In last few minutes, lay a slice of cheese on each burger to melt and lay rolls on grill to toast.
Lay 1 lettuce leaf, then a cheeseburger, on bottom of each roll. Set potato cakes on top. Mix mayonnaise and mustard, dollop about 2 tsp. onto roll tops, and set on burgers.
*Available at lavalakelamb.com, larksmeadowfarms.com, and sunvalleymustard.com
Note: Nutritional analysis is per burger.