Move oven rack to top third of oven. Preheat oven to 400°F.
Mix chicken, fennel, and mushrooms in a large roasting pan or rimmed baking sheet. Drizzle with olive oil, 1/4 teaspoon of salt, and 1/4 teaspoon of black pepper. Toss to coat evenly. Roast in top third of oven, turning once or twice, just until chicken is no longer pink and fennel is almost tender, 5 to 8 minutes. Scrape into a 1 1/2-quart shallow baking dish. Sprinkle with chopped scallions.
Reduce oven temperature to 350°F.
Melt butter in a medium saucepan over medium heat. Stir in flour and cook until golden, 2 to 3 minutes. Gradually whisk in milk until smooth. Whisk in remaining 1/2 teaspoon salt and 1/4 teaspoon pepper. Gradually whisk in half-and-half until smooth and bubbly. Cook until thick, like gravy, about 2 minutes. Stir in thyme and pour over chicken mixture.
Grind bread into crumbs in a food processor (you should have about 1 cup crumbs). Add Asiago and pulse until mixed. Sprinkle bread crumb mixture evenly over top and bake until bubbly and golden, about 35 minutes.
Tip: We use half-and-half to give the gratin a nice thick body and extra flavor, but if you want this dish a little lighter, use low-fat milk instead of the half-and-half.