Spicy Squash Salad with Ginger-Lime Dressing

small butternut squash (about 1 pound), peeled, halved, and cut into 3/4-inch pieces

small acorn squash (1 pound), peeled and cut into 3/4-inch wedges

Coarse salt

very thinly sliced shallot

fresh lime juice

safflower oil

to 2 red Thai chiles, thinly sliced (2 teaspoons), ribs and seeds removed if less heat is desired

finely grated peeled fresh ginger (from a 2-inch piece)

packed thinly sliced fresh basil, plus more for serving

packed thinly sliced fresh mint, plus more for serving

mixed lettuces, such as Boston, Bibb, and lolla rossa or baby red leaf, for serving