Arrange a rack in the middle of the oven and heat the oven to 375°F. If using fresh broccoli, cut 10 to 12 ounces broccoli crowns into small, bite-size florets (about 4 cups).

Dice 1 small yellow onion (about 1 cup) and thinly slice 2 medium celery stalks crosswise (about ¾ cup). Grate 8 ounces sharp cheddar cheese on the large holes of a box grater if needed (about 2 cups). Cut 1 pound boneless, skinless chicken into bite-size pieces (about 3 cups).

Melt 2 tablespoons of the unsalted butter in a large, preferably 12-inch, skillet over medium-high heat. Add the onion, celery, and ¾ teaspoon kosher salt. Sauté until tender and golden-brown, 2 to 4 minutes. Add the broccoli and cook until fresh broccoli is bright green and almost tender, or frozen broccoli is thawed, 2 to 3 minutes.

Add the chicken, half of the cheese, 1 (10 ½-ounce) can condensed cream of chicken soup, ½ cup sour cream, 1 cup whole or 2% milk, 1 teaspoon dry mustard powder, and ½ teaspoon garlic powder. Stir to combine.

Transfer to a 9x13-inch baking dish and spread into an even layer. Top with the remaining cheese. Cover the baking dish with aluminum foil.

Bake until the cheese is mostly melted and the edges begin to bubble, 10 to 15 minutes. Meanwhile, place the remaining 4 tablespoons unsalted butter in a medium microwave-safe bowl. Microwave until almost completely melted, 20 to 30 seconds (or melt on the stovetop and transfer to a medium bowl). Place 30 round butter crackers in a large zip-top plastic bag and seal the bag. Use your hands to crush into coarse crumbs (about 1 ½ cups). Add the crackers to the butter and stir to coat.

Uncover the casserole and evenly sprinkle with the cracker crumbs. Return to the oven and bake uncovered until the edges are bubbling and the top is golden-brown, 10 to 15 minutes more. Let cool for 5 minutes before serving.