1. Preheat oven to 350°. Coarsely chop broccoli. Cook broccoli and 1 Tbsp. salt in a Dutch oven in boiling water to cover 3 minutes or until crisp-tender; drain.
2. Melt butter in a medium-size heavy saucepan over low heat; whisk in flour until smooth. Cook, whisking constantly, 1 minute. Gradually whisk in milk; cook over medium heat, whisking constantly, until mixture is thickened and bubbly. Stir in cheeses and 1/4 tsp. salt until melted. Stir in broccoli and chicken.
3. Place in a lightly greased 11- x 7-inch baking dish; sprinkle with crackers.
4. Bake at 350° for 30 minutes or until golden and bubbly.