Preheat the oven to 400 degrees F.
Toss the bread cubes with 1 tablespoon oil and spread evenly in a single layer on a rimmed baking sheet. Bake until the cubes begin to turn light golden around the edges, 10 to 12 minutes (do not over-brown as the bread cubes will continue to brown when baked atop the sausage in the dish). Remove from the oven and set aside.
Heat the remaining oil in a large nonstick skillet over medium-high heat. Add the onion and sausage. Break up the sausage with a wooden spoon and cook until the sausage has cooked through and the onion is soft, 8 to 10 minutes. Add the spinach and cook, stirring frequently, until wilted, about 2 minutes. Add the sun-dried tomatoes and basil.
Season with 1/2 teaspoon salt and 1/4 teaspoon pepper. Spread the mixture evenly into a 13-by-9-by-2-inch glass baking dish or ceramic baking dish. Sprinkle the goat cheese on the mixture and arrange the bread cubes on top. Make 6 slight indentations in the mixture, spacing apart. Cover and chill. This can be prepared 2 hours ahead.
Crack 1 egg into each indentation, making sure to keep the yolks intact. Sprinkle the eggs lightly with salt and freshly ground black pepper.
Bake until the egg whites are set but yolks are still runny, about 15 minutes (or 18 minutes for a fully cooked yolk).