In a small saucepan over medium-high heat, combine the white wine vinegar, shallots, peppercorns and 2 tablespoons of tarragon leaves. Bring the saucepan to a boil and cook until reduced by two-thirds, about 3 minutes. Remove from the heat and strain through a fine mesh sieve into a heatproof bowl. Add the egg yolk and water and whisk to combine. Set over a pan of simmering water and continue to whisk until the egg starts to thicken, about 2 to 3 minutes. Remove the bowl from the heat, and slowly drizzle a little of the clarified butter into the oil, whisking constantly to incorporate. Return the bowl to the heat, whisk again, and when the egg starts to thicken again, continue to add more of the butter to the egg. Remove from the heat periodically to cool the bowl, and return it once cooled slightly. Continue in this on-the-heat, off-the-heat fashion until all of the clarified butter is incorporated. Remove the bowl from the saucepan, and season with the salt, white pepper, and cayenne pepper. Whisk to incorporate, and fold the tomato puree, the tomato concasse, and the 1 tablespoon chopped tarragon into the sauce. Keep warm until ready to serve.
Add 6 cups of vegetable oil to a large saucepan and heat to 350 degrees F. Fry the potatoes until golden brown, 3 to 4 minutes. Transfer with a slotted spoon to a paper-lined plate or small baking sheet and set aside to drain. Season with salt and pepper.
In a large saute pan heat the olive oil. When hot, add the onions and cook until soft, about 3 minutes. Stir in the trout and the garlic and cook, stirring gently, for 1 minute. Add the potatoes and cook, stirring occasionally, for 2 to 3 minutes. Set aside and keep warm.
Bring a pot of salted water to a boil and add the teaspoon of distilled vinegar. Carefully lay the poached eggs in the water and cook for about 30 seconds, just to rewarm the eggs. Remove the eggs from the water with a slotted spoon and transfer to a paper-lined plate. Season with salt and pepper, to taste.
To serve, spoon the hash in the center of 4 plates. Lay a poached egg on top of each plate of hash. Drizzle the sauce over each plate. Garnish with chives.