This is a great time to soak your bamboo skewers if you haven't already done so. If shrimp are still in the shell, use a sharp scissors and cut along the middle back of each shrimp, leaving the tail intact.
Using a sharp paring knife, cut along the dark vein on the back of each shrimp and remove it. Rinse out the back of each shrimp and pat dry with paper towels.
In a food processor or blender, combine onion, olive oil, basil, parsley, garlic, lemon juice, Dijon, dry mustard, and salt and pepper to taste (I like 1 teaspoon salt and 1/2 teaspoon pepper). Process until smooth. Pour into a zipper-top plastic bag.
Add cleaned shrimp and toss to coat. Cover and marinade for 1 hour at room temperature or up to 2 hours in the refrigerator.
To use a gas grill, turn all burners to high, cover, and heat grill until hot, about 10 to 15 minutes. Clean and oil cooking grate. Add 5 or 6 shrimp per skewer. Grill the shrimp until pinkish white and opaque, about 2 to 3 minutes per side or 5 to 7 minutes total.
To broil in the oven, preheat broiler over high heat. Place skewers 3 inches from the broiler and broil until pinkish white and opaque, about 2 to 3 minutes per side or 5 to 7 minutes total.