For the aïoli
1 small clove garlic
1 large egg yolk
2 Tbs. red wine vinegar
1-1/2 tsp. Dijon mustard
1 cup extra-virgin olive oil
1/4 cup chopped fresh basil
1/4 cup chopped fresh flat-leaf parsley
1/4 cup chopped fresh tarragon
1/4 cup thinly sliced fresh chives
For the burgers
1-1/4 lb. ground beef (80% lean), preferably grass-fed
2 Tbs. finely chopped fresh basil
1 tsp. finely chopped fresh marjoram
1 tsp. finely chopped fresh rosemary
Kosher salt and freshly ground black pepper
4 hamburger rolls, preferably brioche
Lettuce and tomato for serving (optional)
Make the aïoli
Peel, mince, and mash the garlic to a paste with a pinch of salt. In a food processor, pulse the egg yolk, vinegar, mustard, garlic paste, and 1/2 tsp. salt until combined. With the machine running, drizzle in the olive oil until the mixture is emulsified. Add the herbs and pulse once or twice to combine. Transfer to a small bowl and season to taste with salt.
Make the burgers
Prepare a medium (350°F) gas or charcoal grill fire.
In a medium bowl, combine the ground beef, basil, marjoram, rosemary, 1 tsp. salt, and a few grinds of pepper. Form into four 1/2-inch-thick patties. Press the center of each patty with 3 fingers to make an indentation.
Grill the burgers, covered, flipping once, until cooked to your liking, about 6 minutes for medium rare (125°F).
Transfer the burgers to a plate and keep warm.
Toast the buns cut side down on the grill. Serve the burgers on the buns, topped with the aïoli, lettuce, and tomato, if using.
Make Ahead Tips
The aïoli will keep, covered and refrigerated, for up to 5 days.