1. Rinse salt cod well under cool running water. Place in a 1 1/2- to 2-quart jar and cover with water; cover jar airtight and chill 1 to 2 days (see notes), draining and replacing water at least six times. Drain fish. In a 1 1/2- to 2-quart pan, cover salt cod with water; bring to a boil, reduce heat, and simmer until fish flakes easily, 5 to 15 minutes. Drain; when cool enough to handle, remove and discard any bones and skin.
2. Meanwhile, in a 10-inch frying pan or 5- to 6-quart pan over high heat, frequently stir onion and thyme in 2 tablespoons butter until onion is browned, 6 to 10 minutes. Scrape into a bowl.
3. Add broth and potatoes to pan; bring to a boil over high heat, reduce heat, cover, and simmer gently until potatoes mash easily when pressed, 10 to 15 minutes. Drain potatoes.
4. In a food processor (at least 1 qt.) or bowl, combine fish, onion mixture, potatoes, cream, mustard, and pepper. Whirl or beat with a mixer until potatoes are mashed; season to taste with salt.
5. In a buttered shallow 1-quart casserole or four buttered 1-cup ramekins, spread cod mixture level (equally in ramekins) with a spatula, then swirl the top with the tip of the spatula.
6. Break about 1 tablespoon butter into tiny pieces and dot cod mixture evenly, lodging pieces in the crevices.
7. Bake in a 400°; oven until golden brown, 20 to 25 minutes. Sprinkle with capers and parsley. Serve hot.