Using a food processor fitted with a slicing attachment, thinly slice the brussels sprouts.
Core and coarsely chop the apples. In a bowl, toss the apples with 2 tablespoons lemon juice.
In a large skillet, heat the olive oil over medium heat. Add the bacon and cook until crisp, about 5 minutes. Using a slotted spoon, transfer the bacon to a plate. Add the garlic to the remaining fat in the pan and cook, stirring, for 1 minute. Whisk in the mustard, remaining 2 tablespoons lemon juice and the vinegar; season to taste with salt and pepper.
In a large bowl, combine the brussels sprouts, apples, pecans and cheese. Toss with the bacon and warm vinaigrette.