Make herbed cream: Warm oil in a small skillet over medium-low heat. Add shallot and garlic; cook, stirring occasionally, until very tender and fragrant, about 2 to 3 minutes. Set aside to cool.
Combine sour cream, mayonnaise, parsley, dill, lemon zest and juice, and capers in the bowl of a small food processor; pulse to chop. Add shallot mixture and blend until smooth. Taste and season with salt and pepper. (Yield: 2/3 cup. You can make herbed cream up to 1 day ahead; cover and refrigerate. Stir before serving.)
Make fish: Season fish all over with salt, pepper and thyme. Wrap a slice of prosciutto around each piece fish so that seam is placed on the bottom. Preheat broiler.
Warm a large cast iron skillet (or other heavy oven-proof skillet) over medium-high heat. Add oil; swirl around bottom of pan. Place fish in skillet and cook until bottom of prosciutto is lightly browned, 3 to 4 minutes.
Place under broiler; cook until fish is just cooked through and prosciutto is lightly browned on top, 1 to 3 minutes (depending on thickness of fish). Serve fish with herbed cream.