1. In a large nonstick frying pan, heat 1 tablespoon of the oil over moderately low heat. Add the onion and cook, stirring occasionally, until the onion is translucent, about 5 minutes. Increase the heat to moderately high and add another tablespoon of the oil to the pan. Add the mushrooms and cook, stirring, until golden brown, 3 to 5 minutes.
2. Reduce the heat to moderate. Add the remaining tablespoon of oil to the pan. Stir in the eggplant and cook, stirring occasionally, for 5 minutes. Add the salt, cayenne, thyme, tomatoes, and broth. Bring to a simmer. Reduce the heat and simmer, covered, until the eggplant is very tender, about 30 minutes.
3. Meanwhile, in a large pot of boiling, salted water, cook the linguine until just done, about 12 minutes. Drain; toss with the sauce.
Wine Recommendation: Pasta almost always brings us back to Italy and her bountiful variety of wines. A Dolcetto d'Alba, light and fruity but with firm tannins, is what you want to perfectly offset the traditional flavors and high acidity of the sauce.
Notes: When sautéing mushrooms, cook them over relatively high heat and in batches to avoid overcrowding the pan. If the heat's too low or the pan too full, the mushrooms will steam in their own liquid instead of browning. It's worth the extra effort to avoid that rubbery, steamed texture--unless you're especially fond of canned mushrooms.