Preheat your oven to 400°F (205°C).
Bring a large pot of generously salted water to a boil. Cook the pasta until it’s al dente, adding the broccoli to the pot during the final minute of cooking. Drain and set aside.
In a large sauté pan, heat the olive oil or butter over medium-high heat. Sauté the onion for 3 minutes, then add the mushrooms and garlic. Cook until the mushrooms soften and release their moisture, about 5 more minutes.
Sprinkle the flour over the sautéed onion mixture and stir constantly for 1 minute to cook out the raw flour taste.
Gradually add the chicken or vegetable stock, stirring to incorporate any lumps, followed by the milk, Dijon mustard, and seasonings (salt and pepper).
Bring the sauce to a gentle simmer, stirring frequently, until it thickens. Remove from heat and stir in 1 cup of shredded cheddar cheese until smooth. Taste and adjust seasoning, if needed.
In a 9 x 13-inch baking dish, combine the pasta, broccoli, chicken, and mushroom sauce. Toss together, ensuring everything is evenly coated.
Spread the mixture into an even layer. Bake uncovered for 15 minutes. Sprinkle the remaining shredded cheddar cheese over the top and return to the oven for another 10 minutes, or until the cheese melts and begins to bubble.
Let the casserole cool slightly. Garnish with freshly cracked black pepper or chopped parsley, if desired, and serve warm.