Preheat oven to 400 degrees with 1 rack in the middle position and 1 rack immediately below it. Toss radicchio and escarole with 2 teaspoons oil, 1/4 teaspoon salt, and 1/4 teaspoon pepper. Arrange in a single layer on a rimmed baking sheet.
Toss pears with 1 teaspoon oil, 2 teaspoons
thyme, 1/4 teaspoon salt, and 1/4 teaspoon pepper. Arrange in a single layer on another rimmed baking sheet.
Roast radicchio mixture on bottom rack
until tender, about 25 minutes, and pears on
middle rack until tender, about 40 minutes.
Meanwhile, bring water to a boil in a medium
saucepan. Add barley. Simmer, covered,
until tender, 22 to 25 minutes.
Whisk vinegar with remaining tablespoon oil and teaspoon thyme, 1/2 teaspoon salt, and 1/4 teaspoon pepper in large bowl. Drain barley, stir into vinaigrette.
Mound Parmesan onto a baking sheet
lined with parchment or a nonstick baking
mat, and spread into a 6-inch circle.
Transfer radicchio-escarole mixture to
bowl with barley. Cover with plastic wrap to
keep warm and to allow radicchio to wilt.
As pears continue to cook, bake Parmesan
on bottom oven rack until deep golden
brown, 9 to 11 minutes.
Toss pears with barley. Divide salad among
4 plates. Break Parmesan crisp into pieces;
place on top of each salad. Serve warm.