Combine the water, thyme, bay leaf, whole garlic clove, salt, and baking soda in a large saucepan. Bring to a boil over high heat, then stir in the wild rice. Reduce the heat to low, cover, and simmer until the rice is tender and most grains have burst open, 35 to 45 minutes. Set aside without draining.

Melt the butter in a large Dutch oven over medium-high heat. Add the cremini mushrooms, onion, minced garlic, and a pinch of salt. Cook, stirring occasionally, until the mushrooms are well-browned and the liquid they released has evaporated, 12 to 15 minutes.

Stir in the tomato paste and cook until it darkens and begins to stick to the bottom of the pot. Add the sherry and ground shiitakes, scraping up any browned bits from the bottom. Simmer until the sherry has almost entirely evaporated.

Pour the vegetable broth, soy sauce, and the cooked rice along with its cooking liquid into the Dutch oven. Bring to a simmer and cook to allow the flavors to meld, about 20 minutes.

Whisk the cornstarch and heavy cream in a small bowl until no lumps remain. Slowly stir the cream mixture into the soup and continue to simmer until the broth thickens slightly and develops a velvety texture, about 2 minutes.

Discard the thyme sprig, bay leaf, and whole garlic clove. Stir in the chives and lemon zest, then season with salt and pepper to taste. Serve hot.