Preheat oven to 400F/200C (unless you are preparing the dish ahead and plan to bake it later.) Trim the stem ends from the brussels sprouts and cut each one into halves or quarters (depending on how big each one is.) Put the cut brussels sprouts into a salad spinner or colander and wash well with cold water. Spin dry very well (or let drain and blot dry with paper towels if you don't have a salad spinner.)

Put dried brussels sprouts into a bowl and toss with 1 tablespoon olive oil, dried thyme, and a little freshly ground black pepper.

Use a hand grater to finely grate the swiss cheese until you have about a cup (not packed). Mix together the light mayo, mayo, lemon juice, Dijon, and a few grinds of black pepper, then stir in the grated cheese to make the gratin topping.

Mist or brush a large non-stick frying pan with olive oil and heat over medium-high heat until it feels hot when you hold your hand there. Add the brussels sprouts and cook, stirring occasionally, until sprouts have turned bright green and started to brown slightly, about 6 minutes. (Don't overcook at this point; they will cook more in the oven.)

Put sprouts into a gratin dish and spread into a flat layer. Put spoonfuls of the gratin topping in several spots over the top; then use a rubber scraper to spread the topping out into a thin layer. (Don't put it all in the middle and try to spread out; you'll never get it spread out that much.) Sprinkle the coarsely-grated Parmesan over the top. (At this point the dish can be put in the refrigerator to be baked later. I would let it come to room temperature before you bake it, and bake slightly longer at a slightly lower temperature to be sure the center gets hot before the cheese topping is done.)

Put the dish into the preheated oven and bake 20-25 minutes, or until the top is nicely browned and the mixture is heated through. Serve hot.