Place 10 garlic cloves in a saucepan and cover with cold water. Bring to a simmer over medium-high heat and cook 3 minutes, then drain. Return the garlic to the saucepan and cover with fresh water. Bring to a simmer and cook 3 more minutes, then drain again.
Mince the remaining 2 garlic cloves; sprinkle with salt. Use the flat side of a chef's knife to mash the garlic and salt into a smooth paste. (For a stronger aioli, cook 8 cloves in step 1 and mash 4 into a paste.)
Combine the cooked garlic and garlic paste in a food processor. Add the egg yolks and 3 tablespoons warm water; pulse to blend, scraping down the sides as needed. With the motor running, add the olive oil in a steady stream and blend until thick. Transfer to a bowl; cover and refrigerate 1 to 4 hours.
Remove the aioli from the refrigerator about 15 minutes before serving. Whisk in up to 4 tablespoons water to thin it, if desired. Season with salt. Serve with the roasted vegetables.
Choose several vegetables and make a foil packet for each: Pile the vegetables on foil, sprinkle with salt and drizzle with 1 tablespoon olive oil and 2 tablespoons water. Seal into a packet, put on a baking sheet and roast at 450 degrees F until tender. Keep sealed until ready to serve.
Photograph by Anna Williams
Cauliflower 1 head, cut into florets: 30 minutes Turnips 1 bunch, cut into wedges: 30 minutes Fennel 1 bulb, cut into wedges: 30 minutes Baby Carrots 2 bunches, trimmed: 30 minutes Beets 2 bunches, trimmed: 45 minutes to 1 hour Baby Red Potatoes 1 pound: 45 minutes to 1 hour Sweet Potatoes 3 large, cut into wedges: 45 minutes to 1 hour
Courtesy of Food Network Magazine