Heat the oven to 200°C / 400°F. Line a sheet pan (33 x 46 cm / 13 x 18 inch) with foil or parchment for easy cleanup.
Bring salted water to a boil in a 20 cm / 8-inch saucepan. Add the halved baby potatoes and parboil for 8 minutes, until just tender at the edges but still firm in the middle. Drain well.
Spread the potatoes and sausages on the sheet pan. Drizzle with 2 tablespoons of olive oil, scatter over half the garlic and 1 teaspoon of oregano, and toss with your hands. Roast for 20 minutes.
Take the pan out and add the peppers, red onion, remaining garlic and the rest of the oregano. Toss everything together and spread back into a single layer. Roast for 15 minutes more, until the sausages are browned and the vegetables start to char at the edges.
Add the cherry tomatoes and tuck the halloumi slices among the vegetables. Drizzle the halloumi with the remaining tablespoon of olive oil. Roast for 10 to 12 minutes, until the halloumi is golden at the edges.
Check a sausage with a knife or instant-read thermometer. It should read 160°F / 71°C in the center, with no pink and clear juices.
Pull the tray out, squeeze the lemon juice over everything, and scatter with chopped parsley and black pepper.
Serve straight from the pan while everything is warm, with lemon wedges on the side.