Once the garlic has been roasting for about 15 minutes, drizzle the veggies with EVOO and season them with some salt and freshly ground black pepper. Roast them in the oven until tender and caramelized, about 30 minutes.
Once the veggies have been in the oven about 10 minutes, place the tomatoes onto another baking sheet, drizzle them with EVOO and hit them with some salt and freshly ground black pepper. Toss them into the oven with the other veggies and roast until they've burst and shriveled up slightly, 15-20 minutes.
Once the veggies have about 10 minutes left to go, drop the pasta into the boiling water and cook to al dente according to package directions. Before draining the pasta, ladle about 3/4 cup or so of the pasta cooking liquid into a large serving bowl. Drain the pasta and reserve.
Once the garlic is done roasting, allow it to cool enough to handle, then hold the head in your hand over the bowl with the pasta water and squeeze out all of the roasted cloves into the bowl. Add the tomatoes and mash everything together.
Add the roasted veggies, drained pasta, parsley and grated Parmigiano to the bowl and toss it all to combine.
Pass some additional Parmigiano-Reggiano at the table along with the bowl of pasta.