Heat the oven to 375 degrees F.
Place the garlic onto a sheet of aluminum foil, top it with the thyme, drizzle it with 1 tablespoon of the olive oil, and season it with salt and pepper. Close the foil up and roast the garlic until it is soft, about 35 to 40 minutes. When it is cool enough to handle, squeeze the soft pulp into a bowl and set aside.
In a non-reactive bowl or in a food processor, combine the egg yolks, mustard, water, and salt and pepper. Whisk or pulse with the machine to break up the yolks. If you're making the mayonnaise by hand, put the bowl on a damp towel to keep it from moving around while you work. Then drizzle in the oils, whisking constantly, to form an emulsion. If the mayonnaise breaks, stop drizzling and whisk until it comes together again. If you're using the food processor, pour in the oils in a thin stream with the machine running. Then whisk or process in the garlic and lemon juice. Taste and adjust seasoning with more lemon juice, salt, or pepper. Thin the mayonnaise with more water if it is too thick.
Pour the vegetable oil into a deep fat fryer or heavy saucepan to reach halfway up the sides and heat to 325 degrees F. While the oil is heating, peel the skin off the potatoes and cut them into uniform 1/4-inch sticks using a knife, mandoline, or French fry cutter. Rinse the potato sticks well and dry them thoroughly to keep the hot oil from splattering. Fry the potatoes in batches so the pot isn't crowded and the oil temperature does not drop too much. Cook the potatoes until soft but not browned, about 6 to 8 minutes. Remove them and drain on paper towels.
Increase the oil temperature to 375 degrees F. Cook the par-fried potatoes in batches a second time until golden and crispy, about 2 to 4 minutes. Drain on fresh paper towels. Salt the French fries while they are still hot and