Sausage:

Heat grill to medium. Combine all ingredients in a bowl, being careful not to overmix. Divide the sausage mixture into 8 (4-ounce) patties. Grill until cooked completely through, about 4 minutes per side.

Eggs:

Heat grill to medium. Whisk together eggs, salt, pepper, and chives in a medium bowl. Melt the butter in a 9-inch cast iron skillet on the grates of the grill. Add the eggs and stir until soft curds form. Remove from the heat.

Assemble:

Place bagels, cut side down, on the grill and toast until lightly golden brown, about 45 to 60 seconds. Remove the bagel tops from the grill. Turn the bagel bottoms over and top each with a slice of cheese and let melt, about 30 seconds. Top the cheese with a sausage patty and then divide the eggs among the patties. Top with the bagel tops, close the cover and just let heat through, about 1 minute. Remove from heat and serve immediately.