At Bar Tartine in San Francisco, this delicate, delicious oil finishes all sorts of dishes, from squash purée to custards, salads, and soups. You'll need a crockpot with a "warm" setting to slowly extract the flavor of the sunchokes without cooking them.
Coarsely grate sunchokes in a food processor and combine with oil in a crockpot. Cover. Make sure the lid fits tightly; if it doesn't, wrap with foil to make a tight seal.
Turn the setting to "warm" and heat oil until aromatic and infused with the flavor of sunchokes, 10 to 12 hours. Let cool to room temperature.
Line a fine-mesh sieve with cheesecloth and set over a small bowl. Pour oil through cheesecloth and discard solids (don't press them).
Transfer oil to an airtight container, leaving any dark sediment in bowl, and chill. Use cold.
Make ahead: Up to 1 month, chilled.