1. Melt 1 1/2 tablespoons (22 g) of the butter in a heavy saucepan over medium heat. Add the carrot cubes and onion slices and cook, stirring occasionally, for 3 minutes, until the onion is softened and translucent at the edges but not browned.
2. Add the White Stock III and mace blades. Bring to a boil, then reduce heat slightly and boil for 15 minutes, until the vegetables are fully tender and the stock is fragrant with mace.
3. Strain the soup through a fine-mesh sieve into a clean heavy saucepan, discarding the solids. Add the milk (quantity as specified in White Stock III or to desired consistency — see notes).
4. Prepare a beurre manie (a raw thickener made by working equal parts soft butter and flour together into a smooth paste): combine the remaining 1 1/2 tablespoons (22 g) butter with the 3/4 cup (90 g) flour and work together until a cohesive paste forms. Whisk this paste into the hot strained soup in small pieces over medium heat, stirring constantly, and cook for 5 to 8 minutes, until the soup thickens enough to coat the back of a spoon and no raw-flour taste remains.
5. Season with 1 teaspoon salt and 1/8 teaspoon pepper. Remove the saucepan from the heat or reduce to the lowest setting.
6. Stir in the 1/2 cup (50 g) grated mild cheese. Continue stirring for 1 to 2 minutes, until the cheese is fully melted and the soup is smooth and glossy. Serve immediately — do not allow the soup to boil after the cheese is added, as boiling will cause the cheese to become stringy and the soup to break.
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