Preheat your grill and position a rack 6 inches from the heat.
In a 10-inch ovenproof frying pan , cook the bacon over moderately high heat, stirring occasionally, until darker in color and almost fully crisp, about 5 minutes.
Add the shallots and cook, stirring occasionally, until softened, about 3 minutes. Add the brussels sprouts, season with salt and pepper and cook, tossing occasionally, until crisp-tender and lightly browned, about 5 minutes.
Meanwhile, in a large bowl, beat the eggs with the milk, 1/2 to 1 teaspoon of salt (taste a piece of bacon; if it’s really salty, you might be fine with 1/2 teaspoon) and 1/4 teaspoon of pepper. Stir in the shredded cheese and half the chives.
Pour the egg mixture into the skillet and cook over moderate heat, stirring gently, until the eggs start to set, about 1 minute, and then a few minutes more, until the bottom is lightly browned, about 4 more minutes.
Transfer to the grill and cook until the top is a deep golden brown and the eggs are just set in the center, about 5 minutes, but keep an eye on it.
Poke a knife in a couple places to see if any runny egg spills out; if it’s still runny, return it to the broiler for another 1 to 2 minutes. Scatter remaining chives on top and finish with a few grinds of pepper.
If you like, run a rubber spatula around the edge of the frittata and slide it onto a serving plate, then cut it into 6 pieces and serve. Or, you can serve it directly from the pan.