If it's convenient, set the serving plates to warm in a low oven (175 degrees F) as you prepare the fish.
For the sauce, melt 1 tablespoon of the butter in a small saucepan over medium heat. Add the garlic and thyme and cook them briefly until lightly browned. Add the cream, bring it to a boil and reduce quickly by half. Add the remaining 10 tablespoons butter gradually, pat by pat, whisking until incorporated. When the sauce is smooth and thick, discard the garlic and thyme and add the chives, sherry vinegar, tarragon and salt and pepper to taste.
Heat two large cast-iron skillets over medium-high heat. Sprinkle both sides of the fish with salt and pepper and dust the skin sides with the flour. When the pans are hot, add enough canola oil to thinly coat the bottom of each and add a fish to each pan, skin-side down. Cook until the skins turn golden brown and opaque doneness creeps into the pink flesh of the fish. You want to cook the fish nearly all the way through on the skin side. Flip the fish and transfer them to a plate to keep warm in the oven. Cook the remaining fish.
Pour 3 to 4 tablespoons of sauce onto each warm plate, top with a trout, skin side pointing up, and serve immediately.
Recipe courtesy of Amy Thielen, reprinted from The New Midwestern Table. Copyright 2013 by Amy Thielen. Photographs 2013 by Jennifer May. Published by Clarkson Potter, a division of Random House LLC.