Combine 1 Tablespoon water, dairy-free sour cream, 1/2 teaspoon oil, tarragon, and lemon zest and juice in bowl. Season with salt 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 1 teaspoon salt and cook 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 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 remaining 3 cups water, broth, and rice, scraping up any browned bits. Increase heat to high and bring to a 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 spinach or arugula until wilted. Working in batches, process soup in blender until smooth, about 1 minute, or blend with an immersion blender. Return pureed soup to clean pot and season with salt and pepper to taste. Serve, drizzling individual bowls with lemon-tarragon cream.