Bring a large pot of salted water to a boil and cook pasta, until al dente. Reserve ⅓ cup pasta water before draining pasta and setting aside.
Add 1 Tbsp. olive oil to a large skillet over medium heat (make sure skillet is large as you'll be adding all ingredients to it!). Once hot, add sliced chicken sausage and cook for about 5 minutes, until sausage is cooked through and browning around edges. Remove sausage to separate bowl or plate and set aside.
Add remaining 1 Tbsp olive oil to skillet, along with broccoli and garlic. Cook until broccoli is bright green, about 5 minutes.
With heat on low, add chicken sausage back to skillet, along with sliced olives, cooked pasta, feta, lemon juice, salt, pepper, and red pepper flakes. Add half the reserved pasta water. Toss well to combine and warm. Add remaining pasta water if desired.