In a large saucepan, cover the potatoes with cold water and bring to a boil. Cook over moderate heat until tender, 30 minutes. Drain and let cool for 20 minutes. Cut the potatoes in half.
Meanwhile, in a food processor, process the mayonnaise, lemon juice, garlic and mustard. With the machine on, add 1/2 cup of the olive oil and the vegetable oil in a thin stream and blend until emulsified. Season with salt and pepper and refrigerate until chilled.
Light a grill. Toss the potatoes with the thyme and the remaining 1/4 cup of olive oil and season with salt and pepper. Grill over a hot fire for about 10 minutes, turning occasionally, until browned and sizzling. Transfer the potatoes to a platter and serve with the aioli.
The aioli can be refrigerated for 3 days. The potatoes can be boiled 6 hours ahead; grill before serving.
Aioli (EYE-O-LEE) is a garlic-flavored mayonnaise from Provence. It makes a wonderful accompaniment to chicken, cold meats and steamed or boiled vegetables. Aioli can also be mixed with Parmesan cheese, spread on toasted baguette slices and broiled quickly to make excellent appetizers or an accompaniment to a simple green salad. Ailoi can be rescued if it becomes too thick by whisking in a bit of hot water, 1 teaspoon at a time.
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