Set oven to 425°F
Bring a large pot of water to a boil, add kosher salt, then cook pasta to al dente, according to the directions on your pasta.
During the last 2 minutes of cooking, add the broccoli to the pot.
Drain pasta and broccoli, pour back into the large pot and mix in the butter.
Meanwhile, heat a small saucepan over low heat.
Add pine nuts, then continue heating until lightly brown and they give off a nice nutty aroma, ~5 minutes. Be sure to frequently shake the pan during this process.
Set pine nuts aside, then pour balsamic vinegar into the saucepan.
Turn the heat up to medium until the vinegar begins to simmer, then reduce the heat to just high enough to maintain a very light simmer.
Cook down the balsamic vinegar until about 3 tablespoon remains, then pour it into the pot with your drained pasta, mixing to combine.
Pour olive oil into the saucepan and place it on the stove over low heat.
Add the garlic and cook, stirring frequently for 1 minute.
Add stock to the pot, then bring to a boil.
Reduce to a simmer and cook for ~5 minutes.
Season the stock with salt and pepper, then pour it over the pasta.
Into the pasta pot add pine nuts and half of the parmesan, then stir to combine.
Lightly oil or spray a 9x13 inch baking dish.
Distribute pasta evenly into the baking dish, then top with the rest of the parmesan.
Bake for 20 minutes.
Let the pasta rest for about 5 minutes before diving in.