How to Make It
Preheat the oven to 375°. Brush the cut sides of the eggplant with 2 tablespoons of olive oil and place cut side down on a rimmed baking sheet. Bake the eggplant for about 45 minutes, until very soft. Let cool, then carefully scoop out the flesh and coarsely chop it. Transfer the eggplant to a medium bowl.
In a small saucepan, combine the vinegar with the water, 2 tablespoons of the tarragon, half of the shallot and two-thirds of the garlic and bring to a boil. Simmer over low heat until nearly evaporated, about 5 minutes. Transfer the mixture to a blender. Add the chicken stock and one-fourth of the eggplant and puree until smooth. Stir the puree into the remaining chopped eggplant and season with salt and pepper.
In a small skillet, toast the pine nuts over moderately high heat, shaking the pan occasionally, until golden, about 3 minutes; transfer to a plate. In the same skillet, heat the remaining 1/4 cup of olive oil. Add the remaining shallot and garlic and cook over low heat until softened, about 2 minutes. Transfer the shallot and garlic to a bowl and let cool. Whisk in the lemon zest, roasted red pepper, capers and lemon juice and season with salt and pepper. Stir in the remaining 1/4 cup of tarragon and the pine nuts.
Preheat a grill. Brush the sardines with olive oil and season both sides generously with salt and pepper. Grill the sardines over high heat, turning them once, until lightly charred, 2 to 3 minutes. Spoon the eggplant puree onto plates and top with the sardines. Spoon the tarragon dressing on top and serve.