Whisk the heavy cream, sour cream, olive oil, lemon zest, lemon juice, and tarragon in a small bowl until smooth. Season with salt and pepper to taste and set aside to allow the flavors to develop.

Heat the remaining olive oil in a large Dutch oven over medium heat. Add the onion, brown sugar, and a pinch of salt, then cook, stirring occasionally, until the onions are softened and deeply browned, about 15 to 20 minutes.

Stir in the mushrooms and cook until they have released their liquid and turned golden brown, about 5 to 7 minutes. Stir in the garlic and cayenne and cook until fragrant.

Pour in the water and broth, using a wooden spoon to scrape up any browned bits from the bottom of the pot. Stir in the rice and bring to a simmer, then reduce the heat to low, cover, and cook until the rice is very tender and falling apart, about 20 minutes.

Increase the heat to medium-high and add the Swiss chard and kale in large handfuls, stirring until each batch wilts before adding more. Cover and simmer until the greens are completely tender, about 10 to 15 minutes.

Remove the pot from the heat and stir in the parsley and arugula until they wilt from the residual heat. Working in batches, process the soup in a blender until completely smooth and vibrant green—hold the lid down firmly with a folded kitchen towel to prevent splashes.

Return the pureed soup to the pot and warm through over low heat. If the soup is too thick, stir in a little extra water to reach your preferred consistency. Season with salt and pepper to taste and serve hot with a drizzle of the lemon-tarragon cream.