Finely chop the onion and garlic. Leave the rosemary sprigs whole for easy removal at the end of cooking.
In a medium saucepan, melt the butter over medium heat.
Add the chopped onion and garlic and sauté until translucent, allowing the flavors to release.
Pour in the red wine and stir to combine, ensuring the flavors meld together.
Add the rosemary sprigs, beef broth, balsamic vinegar, and brown sugar. Stir well to incorporate all ingredients.
Bring the sauce to a boil, then reduce the heat to low and let it simmer for 15-20 minutes, allowing the flavors to develop and the sauce to slightly thicken.
Remove the rosemary sprigs, then strain the sauce through a fine-mesh sieve to achieve a smooth, silky texture.
If you prefer a thicker sauce, continue to simmer over low heat for another 5-10 minutes.
Season with salt and freshly ground black pepper to taste, then serve warm with your favorite meat dishes.
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