Place the dried mushrooms in a small bowl and cover with 1 1/2 cups very hot water. Set aside and let the mushrooms soften.
Place the carrots, onion, bell pepper, and garlic in a food processor. Pulse the vegetables until finely chopped but still chunky. Place the olive oil in a large, heavy skillet over medium-high heat. Add the chopped vegetables, thyme, oregano, salt, and pepper and cook until tender, about 6 minutes. Strain the porcini mushrooms, reserving the porcini mushroom liquid. Add the porcini mushrooms, fresh mushrooms, and tomato paste and continue cooking, stirring to dissolve the tomato paste, until the mushrooms are softened, about 5 minutes. Add the porcini mushroom liquid and red wine. Bring the liquid to a boil, reduce the heat to low, and let the mixture simmer until the liquid is reduced by half, about 10 minutes.
Meanwhile, bring a large pot of salted water to a boil over high heat. Add the pasta and cook until tender but still firm to the bite, stirring occasionally, about 8 to 10 minutes. Place the pasta in a large serving bowl. Toss with the vegetable bolognese and serve.