1. In a heavy saucepan over medium heat, combine the ham, celery, carrot, onion, and butter. Cook, stirring occasionally, for 8-12 minutes, until the butter and vegetable mixture turns a deep golden to light brown color and the vegetables are softened and fragrant — a process called dry-sauteing with butter, which develops fond (browned flavor compounds) in the fat.
2. Add the flour all at once and stir constantly for 1-2 minutes to form a roux (a cooked paste of fat and flour that thickens sauces). The mixture should smell nutty and darken slightly.
3. Gradually add the Brown Stock and stewed strained tomatoes, stirring constantly to incorporate the roux without lumps. Bring to a simmer.
4. Cook at a gentle simmer for 5 minutes, stirring frequently, until the sauce coats the back of a spoon and a finger drawn across the spoon leaves a clean line.
5. Strain the sauce through a fine-mesh sieve, pressing solids to extract all liquid. Discard solids.
6. Season with salt and pepper to taste. Serve immediately or keep warm over very low heat.
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