To make compote: In a large, heavy-bottomed pot or Dutch Oven, combine apples, berries and water. Set to medium-high heat and cook until the mixture is bubbling.
Once compote comes to a boil, reduce heat to low and cover. Cook, stirring often, until apples have cooked down, for 15 to 20 minutes.
Add sugar, lemon or lime juice and spices, cover and simmer for another 5 minutes. Remove berry apple compote from heat and allow it to cool.
Berry apple compote will keep in a sealed container in the refrigerator for 5 days.
To cook pan-seared duck breast: Using a sharp knife, score duck breasts into a crosshatch pattern (deep scoring will result in more rendered duck fat; score less, and you will preserve more of the skin intact).
Season both sides with salt and pepper, more heavily on the skin side.
Transfer duck, skin side down, into a cold pan. Set pan to medium-low heat and cook duck until most of the fat has been rendered, about 15 minutes (duck should register an internal temperature of 125F). Remove the remaining fat.
Turn duck breast over to the other side and cook for 1-2 minutes for medium rare (130F), 3-4 for medium (140F) and 5-6 for well done (155F).
Remove duck breast from pan and set aside to rest until ready to serve. You can also deglaze the pan with wine, chicken stock and orange juice to make a duck jus.
To serve, choose a large cutting board or platter. Slice the duck breast thinly, and arrange in a heart or two rows. Spoon berry apple compote generously overtop the duck slices.