Hold a sharp knife at a 45-degree angle and cut the florets from the broccoli stems into a bowl. Chop remaining stems into small pieces and transfer stems to a separate bowl.
Heat 2 tablespoons olive oil with butter in a skillet over medium heat and cook garlic in the hot oil-butter mixture, stirring often, until garlic is soft but not browned. Sprinkle in 1 teaspoon red pepper flakes and salt. Pour chicken stock into skillet and bring to a simmer. Stir broccoli stems into the stock mixture and simmer until tender, 10 to 12 minutes. Add a little water or stock if mixture starts to dry out.
Stir broccoli florets into skillet and cook broccoli sauce until florets are bright green and slightly tender, 3 to 4 minutes.
Bring a large pot of salted water to a boil and stir in angel hair pasta. Cook, stirring occasionally, until pasta strands are almost tender, about 4 minutes. Drain and transfer to a large pot.
Pour broccoli sauce over pasta, stir with a wooden spoon, and mix in 1/2 cup Parmigiano-Reggiano cheese. Cover pot with a lid and let stand until pasta absorbs excess liquid and finishes cooking, about 2 minutes.
Serve in bowls and sprinkle each serving with a dusting of Parmigiano-Reggiano cheese and a pinch of red pepper flakes.
