Combine the cream, sour cream, oil, lemon zest, lemon juice, tarragon, and salt in a bowl. Cover and refrigerate to allow the flavors to infuse until the soup is ready to serve.

Heat the remaining oil in a Dutch oven over medium-high heat until it begins to shimmer. Stir in the onion, sugar, and salt and cook, stirring occasionally, until the onion releases its moisture and begins to soften, about 5 minutes.

Reduce the heat to low and cook, stirring often and scraping up any browned bits from the bottom of the pot, until the onion is deeply browned and slightly sticky, about 30 minutes. Adjust the heat as needed; the onions should sizzle gently without scorching, but you may need to increase the flame slightly if they haven't begun to color after 20 minutes.

Stir in the mushrooms and cook until they have released their moisture and become tender, about 5 minutes. Stir in the garlic and cayenne and cook until fragrant, about 30 seconds.

Stir in the water, broth, and rice, scraping up any remaining browned bits from the bottom of the pot. Bring the liquid to a boil, then reduce the heat to low, cover, and simmer for 15 minutes.

Add the chard, kale, and parsley one handful at a time, stirring until each addition is wilted and submerged in the liquid. Return the soup to a simmer, cover, and cook until the greens are completely tender, about 10 minutes.

Remove the pot from the heat and stir in the arugula until wilted. Working in batches, process the soup in a blender until smooth, about 1 minute per batch, taking care not to overfill the blender with hot liquid.

Return the pureed soup to a clean pot and season with salt and pepper to taste. Serve hot, drizzled with the chilled lemon-tarragon cream.