How to Make It
Preheat oven to 450°F. Toss mushrooms and sliced onion with 2 tablespoons oil, ￼½ teaspoon salt, and several grinds of pepper on a rimmed baking sheet. Roast, tossing halfway through, until vegetables are tender and golden, 10 to 12 minutes. Place greens on top of vegetables and return to oven. Roast, tossing halfway through, until greens have wilted, about 5 minutes. Remove from oven, add vinegar, and toss to combine.
Meanwhile, combine hash browns, grated onion, flour, eggs, remaining 1½ teaspoons salt, and several grinds of pepper in a large bowl and mix until evenly combined.
Heat 2 tablespoons oil in a large skillet (preferably cast-iron) over medium-high. Add potato mixture and press into an even layer all the way to the edge; cook, undisturbed, until bottom is golden brown, about 4 minutes; invert onto a plate.
Add remaining 2 tablespoons oil to skillet. Slide latke carefully back into skillet and cook until golden brown on bottom, about 4 minutes. Transfer to a rack. Serve latke, cut into wedges, topped with sour cream and mushroom mixture.