Chop the potatoes and smoked chicken. Add them to a large pot, pour in the chicken stock (or water), and bring to a gentle simmer. Cook over low to medium heat for about 20 minutes, until the potatoes are fork-tender.

While the potatoes cook, finely chop the onions and carrot. Heat oil in a skillet over medium heat. Sauté the onions until soft and fragrant, about 4 minutes, then add the carrot and cook for another 5 to 6 minutes, until softened.

Add the butter and chopped mushrooms to the skillet. Increase heat slightly and cook, stirring often, for about 6 minutes, until the mushrooms release their moisture and turn lightly golden.

Season with salt, paprika, and black pepper. Stir and cook for 1 minute to bloom the spices.

Transfer the sautéed vegetables to the pot with the potatoes and chicken. Add the cream cheese wedges, cover with a lid, and simmer gently for 7 minutes, stirring occasionally, until the cheese is fully melted and the soup is creamy.

Taste and adjust seasoning if needed. Serve hot, garnished with fresh parsley or your favorite toppings.