Bundle the bay leaf, thyme, parsley stems, and peppercorns in a square of cheesecloth, securing the corners with kitchen twine. Ensure one string is long enough to tie to the pot handle for easy retrieval.
In a sturdy medium saucepan, melt butter over medium heat until it turns frothy.
Bring the mixture to a boil, then reduce heat to low, and introduce the sachet. Simmer until the volume reduces by 1/3 (resulting in about 2 cups), stirring often to prevent scorching, for 35 to 45 minutes.
Use a ladle to skim off any impurities that surface.
Remove the sauce from heat, discard the sachet.
For a smoother consistency, carefully strain the sauce through a wire mesh strainer lined with cheesecloth.
Keep the sauce covered and warm until ready to use. If not using immediately, cool it completely and store in an airtight container in the refrigerator for up to 5 days.