In a liquid measuring cup, combine the vinegar, shallots and ½ teaspoon salt. Set aside. Meanwhile, cut off and discard the top third of the garlic head, leaving the head intact. In a 2-quart saucepan over medium-high, combine the garlic, 6 cups water, carrots, celery, mustard seeds, thyme, bay leaves and ½ teaspoon salt. Bring to boil, then cover, reduce heat to low and simmer for 30 minutes. Remove and discard the carrot, celery, bay and thyme.

Return the pot to medium-high and stir in the lentils. Return to a boil, then cover and reduce heat to low. Simmer until the lentils are tender but still hold their shape, 30 to 35 minutes. Remove the garlic and set aside.

Drain the lentils, reserving the liquid, and transfer to a large bowl. Stir in the vinegar-shallot mixture and let cool to room temperature.

Squeeze the pulp from the garlic into a bowl and mash with a fork. Stir in ¼ cup of the reserved cooking water, the oil and ¼ teaspoon salt.

Stir the garlic mixture, half of the cheese, the parsley and half the walnuts into the lentils. Taste and season with salt and pepper. Transfer to a platter and top with the remaining cheese and walnuts.