Preheat oven to 375 degrees F (190 degrees C). Place a mixing bowl in the refrigerator to chill.
Place the mangos and apple juice in a sauce pan; simmer over medium-low heat. After 10 minutes, stir 1/4 cup white sugar and the honey into the mangos. Add the strawberries. Reduce heat to low and continue to simmer until fruit softens.
Place the butter into the chilled mixing bowl, and cut into small pieces. Combine butter with the remaining 1/4 cup white sugar, brown sugar, flour, nutmeg, and cinnamon. Use your fingers or two forks to mix the butter into the dry ingredients until the mixture becomes coarse textured and crumbly. Refrigerate until needed.
Pour the mango mixture into the prepared pie crust. Spread the crumb mixture evenly over the top of the fruit.
Bake in preheated oven until topping is golden brown, about 20 minutes. Remove from oven and cool at least 30 minutes before serving.