Place the Brussels sprouts, pomegranate seeds, and apple slices in a large bowl.
In a small bowl, or container, combine the olive oil, lemon juice, mustard, mayonnaise, honey, shallot and salt and pepper.
Pour the dressing over the salad and toss to combine well. Transfer to a serving bowl.
Notes
The base is slivered Brussels sprouts, which are then roughly chopped. You can easily buy thinly sliced Brussels sprouts, and then just give them a rough chop with a heavy knife on a cutting board. I started with whole Brussels sprouts, gave them a run through the slicing blade of the food processor, and then a rough chop with a knife. Great texture. (I wouldn’t pulse them in the food processor with the regular blade, or use the chopping blade, because I think that would result in too fine a chop and you’d lose the appeal of this nubbbly texture).