Poach the chicken: Melt the butter over medium heat in a wide, shallow Dutch oven or sauté pan. Sauté the onion and garlic in the butter, then add the white wine, if using, and the chicken breasts. Add enough water to cover the chicken breasts by 1 inch. Bring to a boil. Cover and reduce the heat to a bare simmer. Cook for 8 to 10 minutes or until the chicken is cooked through. Cool thoroughly.
Shred with two forks then chop the resulting shreds finely. You should have about 2 1/2 cups of shredded chicken.
Make the salad: Whisk together the yogurt and mayonnaise. Toss with the chicken. Mix in the shallots and dill. Taste and season with salt and pepper.
Refrigerate for at least an hour to let the flavors combine. The chicken salad will keep up to 3 days in the refrigerator.
Substitute cooked chicken: A shortcut for this recipe is buying cooked chicken, like Costco's cooked shredded chicken meat, or using your own precooked chicken from the freezer. Simply substitute about 2 1/2 to 3 cups shredded chicken.