Bring a large pot of generously salted water to a rolling boil. Add the beans and cook until crisp-tender, about 5 minutes. Drain and transfer to a large bowl. Toss with 3 tablespoons olive oil and 1 1/2 teaspoons salt. Cool, uncovered.
Finely grate the zest of 1 lemon; combine with the shallots in another large bowl. Squeeze the juice from both lemons on top. Add the mushrooms and toss with 1 tablespoon salt. Set aside.
Toast the walnuts in a dry skillet over medium heat until golden and fragrant. Sprinkle with the sugar and a pinch of salt and cook until the sugar melts, about 3 more minutes. Remove from the heat and cool.
Add the remaining 1 tablespoon oil to the mushrooms and gently combine with the beans, parsley and half the nuts. Season with salt and pepper. Transfer to a serving bowl or platter and sprinkle the remaining nuts on top.