Arrange a rack in the middle of the oven and heat to 400°F.
Heat 1 tablespoon of the oil in a 10- to 12-inch cast iron or oven-safe skillet over medium-high heat until shimmering. Add the sausages and cook, breaking them up with a wooden spoon, until browned and cooked through, about 10 minutes. Transfer to a paper towel-lined plate and drain any excess fat in the pan.
Add the remaining 1 tablespoon of oil, eggplant, and onion to the pan. Cook, stirring occasionally, until softened, 6 to 8 minutes. Add the garlic, cooked sausage, a big pinch of salt, and stir to combine.
Whisk the eggs and basil together in a large bowl. Pour over the vegetables and sausage. Tilt the pan to make sure the eggs settle evenly over all the vegetables. Crumble the goat cheese evenly over the top. Cook until the eggs at the edge of the pan begin to set, 1 to 2 minutes.
Transfer the pan to the oven and bake until the eggs are set, 8 to 10 minutes. To check, cut a small slit in the center of the frittata. If raw eggs run into the cut, bake for another few minutes. If not, then the eggs are set. Remove from the oven and let cool for 5 minutes. Cut into wedges and serve.
