Prep: Preheat oven to 425°F (218ºC). Line a large 15x21-inch sheet pan (or two smaller pans) with parchment paper.
Parcook potatoes: Place halved potatoes (2 cups) in a microwave-safe bowl with 1–2 tablespoons water. Cover and microwave for 5–6 minutes, until just fork-tender. Drain and set aside.
Make garlic beggar’s purse: Place whole peeled garlic cloves (8-10) in the center of a small piece of foil (about 6x6 inches). Drizzle with 1 teaspoon olive oil and toss to coat. Gather the foil corners together and twist to form a small pouch. Keep it loosely closed so steam can circulate.
First roast (15 minutes): On the prepared sheet pan, spread potatoes, carrots (1 cup), green beans (2 cups), red onion (1 cup), and sausage (13 ounces) in a single layer. Drizzle with 5 tablespoons olive oil, sprinkle with paprika (1 teaspoon), Italian seasoning (1 tablespoon), red pepper flakes (1/4 teaspoon), salt, and pepper. Toss well. Place the garlic beggar’s purse in a corner of the sheet pan. Roast for 15 minutes.
Prep Broccoli + Red Pepper: Meanwhile, toss the broccoli (1 cup) and red pepper (1 cup) with 1 tablespoon canola oil and 1/4 teaspoon salt.
Second Roast: Remove the pan from the oven. Increase oven to 450ºF (232ºC). Scoot the cooked veggies and sausage to the bottom 2/3's of the pan. Add broccoli and bell peppers to the top 1/3 of the pan. Return to the oven for another 7–10 minutes, or until veggies are tender and golden brown in spots.
Finish + Serve: Carefully open the garlic beggar’s purse. The cloves should be soft and golden — mash lightly with a fork if desired, then mix into the veggies or serve alongside. I serve it whole and let everyone mash and use it as they like.
Sprinkle with feta and parsley, if using. Serve hot as-is, or over rice, buttered noodles, or quinoa.