Heat the oven to 400°F (205°C). Grease a 9-by-13-inch, 3-quart baking dish with 1 teaspoon of the olive oil. On a large rimmed sheet pan, toss the peeled 1/2-inch sweet potato cubes with the remaining 1 tablespoon oil and spread in one layer.

Roast for 20–25 minutes, turning halfway, until a fork enters easily and some edges are browned. Leave a little space between the cubes; use two pans if necessary. Reduce the oven to 350°F (175°C) after roasting.

Meanwhile, place the sausage in a large skillet over medium heat. Break into small crumbles and cook for about 8–10 minutes, or until fully cooked and the thickest pieces reach 160°F (71°C). Transfer to a bowl with a slotted spoon. Leave about 1 teaspoon rendered fat in the skillet; drain excess.

Add the diced onion and bell pepper to the skillet and cook over medium heat for 5–7 minutes, stirring, until softened and no liquid pools in the pan. Add the garlic and smoked paprika and stir for 30 seconds. Remove from the heat.

Distribute the roasted sweet potato, cooked sausage and vegetable mixture evenly in the prepared baking dish. Let them stop steaming heavily while you whisk the eggs, milk, salt and black pepper in a large bowl. Pour the egg mixture evenly over the filling. Nudge the ingredients gently with a spatula so the eggs settle between them.

Bake uncovered at 350°F (175°C) for about 35–45 minutes, until the center is set and a food thermometer inserted into the middle registers 165°F (74°C). If the top browns before the middle is ready, cover loosely with foil. Use the temperature and set center rather than the clock alone.

Rest for 10 minutes, then cut into eight portions. Refrigerate leftovers in shallow containers within 2 hours, or 1 hour if above 90°F. Keep at 40°F (4°C) or below and use within 3–4 days. Reheat each portion to 165°F (74°C). Freeze extra portions for up to 2 months for best quality; thaw overnight in the refrigerator before reheating.