Preheat oven to 425 degrees. On a rimmed baking sheet, toss broccoli, oil, 1/2 teaspoon salt, and 1/8 teaspoon pepper. Roast until broccoli is tender and lightly browned, 15 to 20 minutes, tossing halfway through. Meanwhile, in a large pot of boiling salted water, cook noodles until al dente. Reserve 2 cups pasta water; drain pasta.
Return empty pasta pot to medium-low heat. Add cream cheese and 1 cup pasta water; whisk until smooth. Add broccoli and ham and cook until hot. Gently fold in noodles. Thin sauce to desired consistency with more pasta water and season pasta with salt and pepper.
For a shortcut to a pasta supper, make a silky sauce using cream cheese.