1. Preheat oven to 400°F. On a baking sheet, drizzle broccolini with 2 Tbsp. oil; sprinkle with 1/2 tsp. salt and toss to coat. Spread in a single layer. Roast, turning once with tongs, until crisp-tender, 10 to 15 minutes. If stems are not uniform in size, remove thinner ones to a platter as they are done. (Broccolini can be cooked several hours in advance and kept at room temperature.)
2. In a large, deep frying pan, heat remaining 2 Tbsp. oil over medium heat. Add onion and cook, stirring occasionally, until starting to soften, about 5 minutes. Raise heat to medium-high, add mushrooms and remaining 1/2 tsp. salt and cook, stirring occasionally, until mushrooms are golden brown, 7 to 10 minutes. (Mushrooms will release liquid before reabsorbing it and browning.) Add wine; cook until pan is dry, about 2 minutes. Stir in pepper.
3. Spoon mushrooms over broccolini. Scatter some Parmesan over dish, if desired. Serve warm or at room temperature.