Preheat the oven to 400F. Line a baking sheet with parchment paper.
Add the broccoli, 3 tablespoons oil, and salt and toss to coat. Spread out into an even layer. Make a space in the center for the feta and place the feta there. Drizzle the feta with the remaining 1 tablespoon oil and rub it around.
Bake for 12 minutes or until the broccoli is beginning to char.
In a large pot, cook the pasta according to the package instructions. Reserve at least 1 cup of the pasta water, then drain and transfer the pasta back to the pot (you can drizzle/toss with oil if you want to prevent sticking).
While the broccoli roasts, roll the meat into about 16 tiny meatballs. Add the meatballs to the baking sheet and bake for 10 more minutes.
Immediately add all the sheet pan contents into the pot along with about ½ cup pasta water (to start). Break up the feta into chunks and continue to stir everything together until the feta has broken down and mixed with the pasta water to form a sauce that coats the noodles (this will take a minute or two). Add more pasta water along the way, if needed (I ended up using about 1 cup).
Serve the pasta topped with black pepper. I like to cut the little meatballs in half before digging in.