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PdfCombine yogurt, ½ teaspoon oil, tarragon, and lemon zest and juice in bowl. Season with salt and pepper to
taste, cover, and refrigerate until ready to serve.
Heat remaining 2 tablespoons oil in Dutch oven over medium-high heat until shimmering. Stir in onion, sugar, and ½ teaspoon salt and cook, stirring occasionally, until onion releases some moisture, about 5 minutes. Reduce heat to low and cook, stirring often and scraping up any browned bits, until onion is deeply browned and slightly sticky, about 30 minutes. (If onion is sizzling or scorching, reduce heat. If onion is not browning after 15 to 20 minutes, increase heat.)
Stir in mushrooms and cook until they have released their moisture, about 5 minutes. Stir in garlic and cayenne and cook until fragrant, about 30 seconds. Stir in water, broth, and rice, scraping up any browned bits, and bring to boil. Reduce heat to low, cover, and simmer for 15 minutes.
Stir in chard, kale, and parsley, 1 handful at a time, until wilted and submerged in liquid. Return to simmer, cover, and cook until greens are tender, about 10 minutes.
Off heat, stir in arugula until wilted. Working in batches, process soup in blender until smooth, about 1 minute. Return pureed soup to clean pot and season with salt and pepper to taste. Drizzle individual portions with lemon-tarragon yogurt, and serve.