Make a bouquet garni by cutting out a 7-inch square of cheesecloth, placing the bay leaf, thyme and peppercorns in the center, then gathering up the corners. Use a length of cotton kitchen twine to tie the bundle closed.
Place the bouquet garni, lentils, and water in a medium (3 to 4-quart) saucepan over high heat. When the water begins to boil, turn the heat down to low and let simmer, covered, for 25 minutes, or until the lentils are tender.
While the lentils are simmering, begin making the black garlic sauce. In a small (1-quart) saucepan, heat the butter and chopped black garlic over medium heat, just until the butter is melted and begins to bubble. Turn off the heat, then whisk in the mustard, vinegar, and salt.
Discard the carrots, onion, and bouquet garni. Ladle 1/4 cup of their cooking liquid into a wide-mouthed pint mason jar and set aside.
Drain the lentils of all of the remaining cooking liquid, then return them to the cooking pot and set aside, covered to keep warm.
Use a small spatula to scrape every last drop of the garlic/butter mixture from the small saucepan into the mason jar with the 1/4 cup of cooking liquid.
Insert an immersion blender into the liquid in the mason jar, then process until emulsified and smooth. Taste for seasoning and add more mustard, vinegar, or salt if desired.
Pour the black garlic sauce into the pot of lentils. Stir to combine, tasting for seasoning one last time and adjusting if necessary. Serve warm with roast chicken, pork, or a pan-seared salmon fillet.