Combine broth, wine, whole milk, garlic, shallot, and bay leaf in a small saucepan over medium heat. Bring to a simmer and cook until reduced by about half, 10–12 minutes.
Heat one tablespoon butter in a medium saucepan over medium heat until foamy. Sprinkle flour over butter and whisk to combine.
Cook, whisking constantly, until mixture is tan, but not brown, 2–4 minutes.
Slowly and gradually whisk reduced liquid into roux, whisking vigorously to incorporate.
Decrease heat to low and remove bay leaf. Gradually whisk butter into sauce 1 tablespoon at a time, allowing each piece to melt before adding another (this is called mounting the sauce).
Remove from heat and whisk in 1 teaspoon lemon zest, 1 tablespoon lemon juice, and vinegar; season with ½ teaspoon each salt and pepper. Stir in chives and serve.
Optional: thin sauce to desired consistency with warm broth as needed.