Preheat the oven to 450°F or 425°F convection. Line a large rimmed baking sheet with parchment paper.
Add the potatoes to the baking sheet. Drizzle with 1 tablespoon olive oil and sprinkle with the oregano, garlic powder, ½ teaspoon salt, and some black pepper. Toss thoroughly and spread into an even layer. Bake for 15 minutes.
Remove from the oven. Tear the sausage into ping-pong-ball-size chunks (or roll into balls) and nestle them among the potatoes. Scatter the tomatoes all over and sprinkle them all over with ¼ teaspoon salt and drizzle with a touch of olive oil, rolling the tomatoes in the oil to make sure they have a light coating.
Bake for another 20 minutes or until the sausage is cooked through, the potatoes are tender and golden, and the tomatoes have begun to burst.
Add the dressing ingredients to a jar with a lid and shake vigorously to combine.
Add arugula to your bowl along with the sausage, potatoes, and tomatoes, scooping any tomato juices from the pan if possible. Drizzle with some of the dressing and toss to combine. Top with Parmesan and fresh basil.