Crush gingersnaps in a food processor; process until finely ground (makes about 1 cup). Pour in melted butter; process until well combined.
Insert 12 liners in a standard-size square or round muffin pan. Press 2 Tbsp. of the ground gingersnaps firmly into each liner.
Beat goat cheese and cream cheese with a hand mixer on low until smooth. Add 2/3 cup of the sugar and the salt; beat until combined. Incorporate eggs one at a time, then add vanilla. Beat on high until smooth, about 2 minutes.
Pour evenly into muffin liners. Bake at 350º for 15 minutes, until set. Cool to room temperature, then refrigerate until cold.
Meanwhile, combine blackberries, raspberries, blueberries, remaining 1 Tbsp. sugar and 1 Tbsp. WATER in a pot. Cook over medium heat until berries burst, about 5 minutes. Cool. Refrigerate until using.
Remove cakes from pan and peel off liners. Spoon berries on top of each cake and garnish with mint leaves if desired.