Preheat the oven to 425°F and line a large, shallow baking pan with parchment paper.
Put a large pot of salted water on to boil for the pasta.
Peel the eggplant and cut it into 3/4-inch cubes. Transfer the cubes to a large bowl and drizzle with 1 to 2 tablespoons of olive oil. Season with salt and pepper and toss to combine.
Arrange the eggplant in a single layer on the prepared baking sheet and roast until tender and lightly caramelized, 15 to 20 minutes. Remove from the oven and set aside.
While the eggplant roasts, heat 2 tablespoons of olive oil in a large skillet over medium heat. Add the onion and cook until soft and translucent, 4 to 5 minutes. Add the garlic and continue cooking until fragrant, 1 minute more.
Add the mushrooms, season with salt and pepper and cook until they're golden brown, 5 to 7 minutes.
Stir in the diced tomatoes, balsamic vinegar, tomato paste and oregano. Taste and adjust the seasoning as needed. Simmer, stirring occasionally, for 5 minutes, then add the eggplant to the pan.
Continue cooking for 7 to 8 minutes longer, stirring occasionally. Stir in the parsley and reduce the heat to low to keep warm.
Cook and drain the pasta according to package directions and return it to the pan. Drizzle with a bit of olive oil and season with salt and pepper.
To serve, divide the pasta between four plates or pasta bowls. Top with the eggplant ragu and some grated Parmigiano-Reggiano.