Combine broth and next 3 ingredients in a saucepan over high heat. Bring to a boil. Cook over high heat 15 to 20 minutes or until reduced by half.
Stir in vinegar. Cook 10 minutes or until mixture has reduced to about 1/2 cup and has the consistency of a light sauce.
Remove from heat, strain sauce through a fine-mesh strainer, and return to pan. Whisk in butter. Season to taste with salt and pepper.
Note: if making ahead, warm gently and whisk in butter just before serving.
Chef John Recommends: Red wines that are lower in tannins but "fruit forward" (meaning fruit is one of the first elements you taste), to enhance the beets. Though he produces noteworthy white wines, he says that grilled rich and oily fish such as salmon are perfect foils for reds. His favorite varietals with this dish are Pinot Noir or, for something different, a French Grenache or Spanish Garnacha.