Clean and trim the sprouts, cut them in half lengthwise, and then slice thinly lengthwise. The thinner you can make the slices, the better.
Place a frying pan on the stove at low medium heat, and melt your coconut oil or butter, and throw in the sliced shallot. Let the shallot cook for a couple of minutes until it's just starting to turn lightly golden.
Add the Brussels sprouts, and stir gently to coat them with the melted oils.
Turn up the heat to medium/medium high. Keep stirring the sprouts so that they don't burn, but just start to turn brown on the edges.
As soon as the sprouts have turned bright green and have caramelized spots on them, transfer to a plate, and sprinkle with salt and pepper to taste.