Combine shallots, vinegar, and wine and reduce by 1/4. Add tomato puree and reduce by 1/2. Pour into a bowl and whisk in the egg yolks. Place over simmering water and whisk until light, airy and ribbony. Slowly whisk in melted butter. Season with salt and pepper.
Preheat the oil. In a saucepan, over medium-high heat, combine the tomatoes, tarragon and 1 tablespoon garlic. Season the mixture with salt and cayenne. Stir in 3 tablespoons vinegar and red wine. Bring to a boil. Reduce the heat to medium and simmer for 20 to 25 minutes, or until the mixture is dark brown and thick. Remove from the heat and add the parsley. Set a stainless steel bowl over a pot of simmering water. Whisk the egg yolks with the juice of the lemon and water. Season the egg mixture with salt and cayenne. Whisk until the eggs are pale yellow in color and slightly thick. Remove the bowl from the pot and whisk in the butter, 1 tablespoon at a time, until all is incorporated. Fold the tomato mixture into the hollandaise and set aside. Fry the potatoes until golden brown, about 3 to 4 minutes. Remove from the oil and drain on a paper-lined plate. Season the potatoes with salt and pepper. In a large saute pan, over medium heat, add the olive oil. When the oil is hot, add the onions and saute for 2 minutes. Stir in the garlic and trout. Continue to saute for 1 minute. Stir in the potatoes and continue to saute for 2 to 3 minutes. Over high heat, bring a pot of salted water to a boil. Add the vinegar to the water. Carefully lay the poached eggs in the water and cook for about 30 seconds, just to re-warm the eggs. Remove the eggs from the water and drain on a paper-lined plate. Season the eggs with salt and pepper. To assemble, spoon the hash in the center of four plates. Lay a poached egg on top of each plate of hash. Drizzle the choron sauce over each plate. Garnish with chives.