Preheat oven to 400°F. Stir together chicken, Broccolini, 1 tablespoon of the oil, ½ teaspoon of the salt, and ¼ teaspoon of the pepper in an 11- x 7-inch (2-quart) baking dish until evenly coated. Bake, uncovered, until chicken and Broccolini are partially cooked, about 10 minutes.
Meanwhile, heat 1 tablespoon of the oil in a large skillet over medium-high. Add onion; cook, stirring often, until tender, about 3 minutes. Add garlic; cook until fragrant, about 30 seconds. Reduce heat to medium, and stir in flour; cook, stirring constantly, until incorporated, about 30 seconds. Gradually whisk in milk until smooth; bring to a simmer over medium. Cook, whisking often, until slightly thickened, 5 to 8 minutes. Remove from heat, and whisk in 1 ½ cups of the fontina, remaining ½ teaspoon salt, and remaining ½ teaspoon pepper until smooth.
Transfer chicken and Broccolini to sauce in skillet, discarding any juices in baking dish. Add rice to skillet; stir until just coated. Spoon mixture into baking dish.
Stir together panko, thyme, remaining 1 tablespoon oil, and remaining ½ cup fontina in a small bowl; sprinkle evenly over chicken mixture. Bake, uncovered, at 400°F until top is golden brown, 25 to 30 minutes. Garnish with additional thyme.
