Make herb salad: Place baby arugula leaves, dill and Italian parsley in a bowl. Toss to combine.
Make caper mayo: Place mayonnaise, sour cream, capers and lemon juice in a bowl. Mix to combine. Cover and refrigerate until ready to serve.
Place a large skillet over medium-high heat. Shen hot, add oil and butter.
Place red snapper fillets on a plate and sprinkle with salt and pepper.
Place the fillets in the pan and cook, turning once, until they are lightly browned and just cooked through, about 3 minutes.
Serve each fillet on a dinner plate, spread liberally with caper mayo and top with herb salad. Serve with buttered jasmine rice and a lemon wedge on the side.