For the vinaigrette, place the shallots in a fine-mesh strainer and rinse with cold water. Drain them and transfer to a small bowl with the garlic, sugar, and salt. Strip the thyme leaves off the stems and finely chop the leaves (for about 1 teaspoon thyme); stir into the bowl. Stir in the vinegar, water, Dijon mustard, and whole grain mustard. Pour the olive oil into the bowl in a slow stream, whisking all the while until emulsified. 

The vinaigrette can be refrigerated for up to 3 days. 

For the salad, pull off any wilted or bruised outer leaves from the butter lettuce. Set aside the floppy, darker green leaves for another use; you will only use the pale inner head. Wash the leaves in two changes of water: First, fill a basin with lukewarm water, and soak the lettuces in it, swishing with your hands. Lift the leaves out and drain in a colander. Second, wash the leaves in cold water, again swishing them with your hands and lifting them out. Rinse the leaves well. Slice the frisee leaves into smaller pieces and separate the Little Gem leaves and remove any tough stems from the cress; wash them in the same way. 

Spin all the leaves dry in a salad spinner, then spread them out on a large, lint-free kitchen towel. In all, you will have about 6 handfuls of mixed leaves. Gently press on them with another towel and roll them up completely. 

Place all the leaves in the largest bowl you can find. Season them with a good pinch of salt and a few grindings of pepper. Drizzle in most of the vinaigrette, tossing with your hands to coat the leaves thoroughly. 

Lay the leaves on a plate in gradual stages, so they can be piled high without falling. Tuck the endive spears around the sides and drizzle with a little bit more vinaigrette.