For the celery root: Place the celery root in a 5-quart saucepan. Fill the pan halfway full with cold water. Add the butter and salt. Bring to a boil over high heat. Cook until tender, 20 to 25 minutes. Drain in a colander.

For the artichokes: Place a grill pan over medium-high heat or preheat a gas or charcoal grill. In a medium bowl, toss together the artichokes, oil, garlic, thyme, salt and pepper. Grill, turning with a spatula occasionally, until lightly charred, 2 to 3 minutes. Set aside to cool slightly.

Place the artichokes in a food processor and blend until smooth. Add the cooked celery root, milk, butter, salt and pepper to the food processor. Blend, scraping down the sides of the bowl as needed, until the celery root is smooth and blends into the artichoke mixture. Transfer the puree to a serving bowl or keep warm over low heat in a large saucepan until ready to serve.