Step 1

Place fish bones and heads in a large bowl and cover with cold water. Stir in kosher salt until dissolved. Let stand 1 hour. Drain, then rinse fish under cold running water, washing away any large areas of blood (such as near spine).

Step 2

In a large saucepan, heat oil over medium-high heat until shimmering. Add onion, fennel, leek, celery, and garlic, and cook, stirring, until vegetables have softened slightly, about 3 minutes. Stir in fish heads and bones. Add wine and cook, stirring, until it begins to steam. Add 4 cups water (liquid should just barely cover heads and bones; if not, add just enough more to barely cover). Add parsley, tarragon, bay leaf, and peppercorns.

Step 3

Bring liquid to a bare simmer, then lower heat so that it stays just below a simmer, with only the occasional bubble. Cook for 20 minutes. Using a spoon, skim off any scum that accumulates on the surface.

Step 4

Strain fish stock through a fine-mesh strainer, then chill. Fish stock can be kept refrigerated, covered, for up to 5 days or frozen for up to 3 months.