Preheat oven to 350 degrees. Brush a 3-quart broiler-proof baking dish with oil. Bring milk to a simmer in a pot. Add chicken; simmer until cooked, 5 minutes. Transfer to a plate. Remove pot from heat; whisk in mustard and cream cheese. Season with salt and pepper.
Shred chicken. Cook noodles in a pot of boiling salted water 3 minutes. Add Broccolini; cook until noodles and Broccolini are al dente, 2 minutes. Reserve 1/2 cup pasta water. Drain noodles and Broccolini; toss with chicken, sauce, pasta water, and cheddar. Transfer to baking dish.
Toss breadcrumbs with oil; sprinkle over casserole. Cover with parchment-lined foil; bake 20 minutes. Uncover casserole; increase heat to broil. Cook until top is golden brown in places and breadcrumbs are crisp, 4 to 5 minutes.