1. In a heavy saucepan (4–5 qt / 3.8–4.7 L), melt the butter over medium heat. Add the diced onion, carrot, celery, and ham. Cook, stirring occasionally, for 5 minutes, until the vegetables are softened and beginning to color at the edges.
2. Add the flour to the pan — this step is called making a roux: the flour coats the fat and will thicken the soup. Stir continuously and cook for 3 minutes, until the mixture smells faintly nutty and no raw-flour aroma remains.
3. Add the canned tomatoes with their liquid. Stir to combine with the roux base. Add the peppercorns, bay leaf, cloves, and thyme. Cover the pan.
4. Cook slowly at a bare simmer over low heat for 1 hour, stirring occasionally, until the tomatoes have broken down completely and the mixture is thick and deep red-orange in color. Alternatively, place the covered pot in a 300°F (150°C) oven for 1 hour — this requires less active watching and produces even, gentle heat.
5. Remove from heat. Discard the bay leaf. Pass the mixture through a fine-mesh strainer or food mill, pressing solids firmly to extract all liquid. Discard the solids. This step clarifies the soup and removes peppercorns, cloves, and fibrous matter.
6. Return the strained soup to a clean saucepan. Stir in the hot brown soup stock. Taste and season with salt and black pepper. Reheat gently over medium-low heat, 3–5 minutes, until steaming and just beginning to simmer — do not boil vigorously once stock is added. Serve immediately.
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