Preheat the pellet grill to 225°F / 107°C. Use its smoke setting if available at that temperature.

Prepare the vegetables. Toss the tomatoes, mushrooms, and peeled garlic with olive oil, pepper, half the basil, and a little salt. Go easy on the salt because the feta and BBQ rub may already contain plenty.

Assemble in a shallow, grill-safe baking dish. Use a metal roasting pan, cast-iron skillet, or sturdy foil tray. Spread the vegetables around the feta block in the center. Drizzle the feta with olive oil and sprinkle with chili flakes and BBQ seasoning. Keep the vegetables fairly spread out so they roast rather than sit in a deep pile.

Smoke uncovered for 30–45 minutes. Keep the lid closed. This stage adds smoke flavor, but the vegetables and cheese will still need the hotter finishing stage.

Increase the grill to 400°F / 204°C. Leave the dish on the grill and cook for another 25–40 minutes, checking occasionally. It is ready when: The tomatoes have burst and released their juices. The mushrooms are tender. The garlic is soft enough to mash. The feta is soft enough to stir into the sauce. It will not melt like mozzarella. If the edges look dry before the garlic softens, stir the vegetables and add a small splash of water.

Cook the pasta separately. Boil in salted water until al dente. Reserve about 1 cup of pasta water before draining.

Make the sauce and toss. Mash the garlic and tomatoes, then stir in the softened feta. Add the cooked pasta and enough reserved pasta water, a splash at a time, to make a creamy sauce that coats the noodles.

Finish and serve. Add the remaining basil, taste before adding more salt, and top with a little Parmesan if desired.