1. Place the butter in a heavy saucepan over medium heat. Add the carrot slice, onion slice, bay leaf, thyme sprig, and parsley sprig. Cook, stirring occasionally, for 6-10 minutes until the butter turns a deep golden-brown (a process of browning aromatics in fat — similar to building a compound roux base). The butter should smell nutty and the vegetables should be softened and lightly caramelized.
2. Remove and discard the carrot, onion, bay leaf, thyme, and parsley. The browned butter with its infused flavor remains in the pan.
3. Add the flour all at once, stirring constantly over medium heat for 3-5 minutes until the flour-butter paste (roux) is well browned — it should deepen to a rich brown color, smell toasted and nutty, and pull away slightly from the sides of the pan.
4. Add the stewed and strained tomatoes and the Brown Stock gradually, whisking or stirring constantly to prevent lumps. This is an incorporation step analogous to deglazing (adding liquid to a hot fat-and-starch base to form a smooth sauce).
5. Continue to cook over medium-low heat, stirring frequently, for 10-15 minutes until the sauce thickens enough to coat the back of a spoon and a line drawn through it with a finger holds its shape. Season to taste with salt and pepper.
6. If desired, strain the finished sauce through a fine-mesh sieve for a smoother result before serving.
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