Melt the butter in a pot and sauté the chicken strips until they turn white. Mix in the flour until well incorporated with the butter, then gradually add the hot water until the pot is halfway filled. Bring to a boil and add the carrots, shallots, and peas, letting them cook until halfway done before adding the mushrooms. Continue boiling until the vegetables and chicken are fully cooked. In a separate bowl, whisk together the egg yolks and lemon juice, then stir in the cream. Gradually mix in some of the hot soup to temper the eggs and cream, then slowly add this mixture back into the soup, stirring constantly to prevent the eggs from curdling. Lower the heat, and stir in the salt, pepper, and parsley, letting it boil for an additional 3-4 minutes before removing from heat.