Position a rack in the center of the oven and heat the oven to 400°F.
Using a serrated knife, carve the crust off the bread into rustic, curved slabs that are about 1/4 inch thick. Save the rest of the bread for another use (such as making breadcrumbs). Brush the crusts on both sides with olive oil and season lightly with salt. Put the crusts on a baking sheet and bake until crisp and golden brown, 6 to 8 minutes. When cool enough to handle, rub the crusts lightly with the cut sides of the garlic clove. Snap the crusts into bite-size pieces. Discard the garlic.
In a small bowl, combine the shallots and vinegar with a pinch of salt and let sit for at least 10 minutes and up to 2 hours.
When ready to serve, put the chapons and lettuce in a large mixing bowl and season with a generous pinch of salt and a few grinds of pepper. Drizzle the lettuce with the 6 Tbs. olive oil. Scoop the shallots out of the vinegar and sprinkle them on the lettuce. Gently toss the salad, making sure that all of the lettuce is evenly dressed. Taste and adjust the seasoning with more olive oil, the remaining vinegar in the bowl (or more if necessary), salt, and pepper. Serve immediately on a chilled platter or individual plates, with the chapons tucked in among the lettuces.