In a cast iron skillet, cook the bacon over medium heat until crispy, about 8 minutes (4 minutes each side). Remove the bacon from the skillet and place to the side. Leave the bacon fat in the skillet.
Return the skillet to medium heat and let the fat return to heat. Cut the bagel in half and lightly soak the flat side of each in the bacon fat. Toast each side of the bagel flat side down in the bacon fat, about 2 minutes each. The flat sides of the bagel halves should be golden brown. If there’s too much bacon fat underneath the bagel for it to properly toast, tilt the skillet so the fat pools to one side. Then, use tongs to place the bagel flat side down above the pooled fat. Alternatively, if the bacon fat smokes too much for your kitchen, lightly soak the flat sides of the bagel halves in bacon fat and toast in a toaster oven (or standard oven) until golden brown. Take both sides of the bagel off heat. Add the grated cheese to one side of the bagel and place in a toaster oven (or oven) at 350 degrees Fahrenheit for 2-3 minutes, or until the cheese is melted to your liking.
Crack two large, cold eggs into a nonstick pan with a tablespoon of cold butter. Over high heat, stir the eggs with a rubber spatula for 30 seconds, then alternate by cooking the eggs off-heat and on-heat every 10 seconds for about 3 minutes, mixing constantly (this technique is my favorite for scrambled eggs — it’s from Gordon Ramsay). As you’re finishing, take the pan off heat and add the chopped chives and cream cheese, as well as salt and pepper to taste. Stir until all the ingredients are fully incorporated. The scrambled eggs should a bit custardy — if you like your scrambled eggs done more, feel free to adjust by cooking longer.
Place the scrambled eggs on the half of the bagel without cheese, place bacon on top, and add the half with melted cheese on top. Serve hot.