Heat oven to 350ºF with a rack in the middle.
Bring 3 quarts of water and 2 tablespoons of the salt to a boil in a large, heavy pot, such as a Dutch oven. Prepare an ice bath in a salad spinner. Add the beans to the boiling water, return to a boil, and cook until just tender, about 8 minutes. Drain the beans, then immediately transfer to the ice bath. Let sit until cool, about 5 minutes, then remove the basket insert and pour out the ice water. Spin the greens to dry them. Dry the pot well.
Next, pour 3/4 inch of oil into the now-dry pot, and bring to 375°F over medium-high heat. Line a half sheet pan or platter pan with paper towels and stash next to the cooktop.
While the oil heats, combine the onions and rice flour in a medium bowl and toss to coat the slices evenly in the flour. When the oil hits its target temp, carefully add about a quarter of the onion slices to the oil and fry, stirring occasionally, until golden brown, about 5 minutes. Use a spider to transfer to the prepared sheet pan and season lightly with salt. Repeat with the remaining onions, then set aside. (The onions can be made up to a week ahead and stored in airtight containment at room temperature.)
Melt the butter in a 12-inch cast iron skillet over medium heat. When the butter is foamy, add the mushrooms, the 2 teaspoons salt, and the pepper. Cook, stirring frequently, until the mushrooms have given off all of their liquid, the liquid has evaporated, and the mushrooms have browned, about 10 minutes
Add the garlic and nutmeg and cook, stirring constantly, until aromatic, about 30 seconds. Sift the flour over the mushrooms, then continue to cook, stirring, until the rawness has been cooked out of the flour, about 1 minute.
Stir in the vinegar and cook about 1 minute. Slowly stir in the broth, scraping up any browned bits from the bottom of the skillet. Bring to a simmer, then stir in the half and half. Return to a simmer, then reduce the heat to low. Continue to simmer, stirring frequently, until the mixture is quite thick, about 12 minutes. Remove from the heat.
Stir in the green beans and half of the fried onions, then top with the remaining onions. Transfer to the oven and bake until the sauce is bubbly around the edges and the onions are deeply browned, about 10 minutes.