Preheat the oven to 350 degrees F.
Melt 1/4 cup of butter in a heavy skillet over medium-high heat. Brown the duck breasts skin side down for about 2 minutes without shaking the pan. Turn over and cook on the other side for another 2 minutes. Remove to a baking sheet and bake in the preheated oven for 10 minutes. Transfer the duck breasts to a cutting board and set aside to rest.
Using the same pan you cooked the duck in, saute the onions, over medium heat, until golden brown. Add in the chopped apricots, apple cider vinegar, chicken stock and white wine. Cook until reduced by half, about 10 minutes.
In a small bowl, mix the cornstarch with a little cold water to create a slurry. Slowly add it to the apricot mixture, stirring constantly until the desired consistency is reached.
Allow to cook until the paste taste disappears, about 3 to 4 minutes. Season with salt and pepper, to taste. Stir in the tarragon and the remaining 3 tablespoons of butter.
Slice the duck breast on a bias. Lay a little sauce on the base of each white serving plate and top with the duck. Spoon a little more sauce over the duck and serve.
This dish would be great with sauteed asparagus and boiled fingerling potatoes.