Roast seven or eight big plump cloves of garlic. Mash them and blend with maybe a quarter-cup of lemon juice, some salt, a little black pepper, some tamari or soy sauce, and four or five eggs. I like to use the blade attachment of a hand blender for this part, so that the roasted garlic gets evenly distributed, and then switch to the whisk attachment for the next part.

The next part is where you add three cups of olive oil, but not all at once. If you've never made mayonnaise before, pay attention; this is the important part.
Drizzle in the oil a little at a time, making sure each addition is completely blended in before adding the next bit. The mixture should be thick; once it's thick, turn the blender off, because if you beat it too long it will thin again.

Store aioli in a tightly covered jar in your refrigerator for up to a week. Serve as a dip with a platter of freshly cooked and raw vegetables -- my personal favorites include baby artichokes (see recipe), red potatoes (boiled until tender but still firm), green beans, asparagus, broccoli (barely steamed -- should all still be bright green and crisp), cherry tomatoes, baby carrots, cucumber spears and mushrooms (all raw). As long as you're dipping stuff, try cheese cubes or hunks of bread.

Aioli also makes an excellent sandwich spread.