In a small bowl or container, combine the mayonnaise, olive oil, mustard, lime juice, orange juice, honey, red onion, salt, and pepper.
Pour the dressing over the Brussels sprouts and toss to combine well. Add the Parmesan, dried cranberries and pumpkin seeds and toss again. Transfer to a serving bowl.
Notes
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 have used the pre-sliced version, but in my more perfect world, I like to start with whole Brussels sprouts, run them into the chute of my food processor, and through the slicing blade. Then, I dump them onto a cutting board and give them a rough chop with a knife.