Stir together ground lamb, shallot, Worcestershire, thyme, 1 tablespoon of the mustard, and 1/4 teaspoon each of the salt and pepper in a large bowl until well combined. Divide evenly into 4 (about 4 1/2-ounce) balls. Shape each ball into a 4-inch-wide, 3/4-inch-thick patty. Arrange patties on a baking sheet lined with parchment paper; cover with plastic wrap. Chill patties at least 30 minutes or up to 8 hours.,
STEP 2
Stir together mayonnaise, sour cream, garlic, 1/4 teaspoon each of the salt and pepper, and remaining 1 tablespoon mustard in a small bowl until well combined. Cover; place in refrigerator until ready to use or up to 3 days.,
STEP 3
Preheat a grill to medium-high (400°F to 450°F). Sprinkle patties evenly with remaining 1/2 teaspoon salt and 1/4 teaspoon pepper. Place patties on oiled grates. Grill, covered, until a thermometer inserted into thickest portion of patties registers 140°F to 145°F for medium doneness, about 5 minutes per side.,
STEP 4
Serve on toasted buns topped with Dijonnaise mixture, feta, onion, and arugula.