Preheat the oven to 400°F. Put the broccolini on a baking sheet, drizzle with 2 tablespoons of the oil, sprinkle with 1/2teaspoon of the salt, and toss to coat evenly, then spread in a single layer. Roast the broccolini, turning once with tongs, for 10 to 15 minutes, until crisp-tender. If the broccolini stems are not uniform in size, remove thinner ones as they are done. Transfer the broccolini to a platter. (The broccolini can be cooked several hours ahead of time and kept at room temperature).
In a large (12 inches or wider), deep frying pan, heat the remaining 2 tablespoons oil over medium heat. Add the onion and cook, stirring occasionally, for about 5 minutes, until starting to soften. Raise the heat to medium-high, add the mushrooms and the remaining 1/2 teaspoon salt, and cook, stirring occasionally, for 7 to 10 minutes, until the mushrooms are golden brown. (The mushrooms will release a lot of liquid before absorbing it and browning. Be patient, as the flavor is in the browning.)Add the wine and cook for about 2 minutes more, until the pan is dry. Stir in the pepper.
Spoon the mushrooms over the broccolini, then scatter some Parmesan over the top. Serve warm or at room temperature.