1. In a large pot, heat 2 tablespoons of the oil over moderate heat. Add the carrots, onions, celery, and garlic. Cook, stirring occasionally, until the vegetables start to soften, about 5 minutes.
2. Add the split peas, parsley, water, thyme, and bay leaf. Bring to a boil. Reduce the heat and cook at a low boil, covered, until the peas are almost tender, about 35 minutes. Add 1 3/4 teaspoons of the salt and simmer 5 minutes longer. Remove the bay leaf.
3. Meanwhile, in a large nonstick frying pan, heat the remaining 2 tablespoons oil over moderately high heat. Add the mushrooms and the remaining 1/4 teaspoon salt. Cook until the mushrooms brown, 5 to 10 minutes. Add the mushrooms to the soup and bring back to a simmer. Stir in the Parmesan and pepper.
Wine Recommendation: You'll need a white wine with body and lots of character to stand up to a hearty pea soup like this one. Greco di Tufo, with its full texture and big, nutty taste, is made for the job.