Preheat oven to 350°. Toss chestnuts in oil and put on a rimmed baking sheet. Bake, turning often, until slightly toasted but not dry, 10 to 15 minutes.
Bring a large pot of salted water to a boil. Add brussels sprouts and cook until bright green and just tender, about 6 minutes. Drain and rinse with cold water.
Cook bacon in a large frying pan over medium heat, stirring often, until almost crisp. Drain bacon in a strainer, reserving fat in a bowl.
Wipe out pan and add 3 tbsp. reserved bacon fat. Add shallot and cook, stirring often, until softened and starting to brown, about 3 minutes. Add drained brussels sprouts and cook, stirring often and adding more fat if needed, until browned and starting to crisp, about 6 minutes. Sprinkle with salt, pepper, nutmeg, and lemon juice, stirring to combine.
Transfer to a serving bowl and top with reserved bacon and chestnuts.