Place all the ingredients in a medium saucepan. Cover with cold water by 1 to 2 inches and bring to a simmer over medium-high heat; reduce heat to medium low and simmer gently until the chicken is completely cooked and no longer pink in the center, about 20 minutes.
Heat the oil and butter in a large frying pan over medium heat until the butter is foaming.
Add the shallots, salt, and pepper and cook, stirring occasionally, until the shallots are softened and just beginning to brown, about 10 minutes.Stir in 1/4 cup of the vinegar and the sugar and cook, stirring occasionally, until the shallots are completely soft and caramelized and the jam is thick and syrupy, about 20 to 25 minutes.
Remove the pan from heat and stir in remaining 2 tablespoons of vinegar.
Let the jam cool for 20 minutes, then stir in the chopped tarragon.To assemble the sandwiches:Slice the chicken into 1/4-inch thick pieces.Evenly divide the shallot jam on 1 side of 4 slices of the bread. Divide the chicken evenly among the sandwiches, placing it on top of the jam in an even layer. Close each sandwich and serve.Beverage pairing: Louis Bouillot Crémant de Bourgogne Rosé, France. A lovely, simple sparkler made from Pinot and Chardonnay in Burgundy, this rosé is light and delicate and will perform flawlessy alongside the chicken. Its pinkish-red fruit and hint of sweetness will also make it a natural partner for the shallot jam.