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Mix flour, ginger, cinnamon, baking soda, nutmeg and salt in large bowl. Set aside. Beat butter and brown sugar in large bowl with electric mixer on medium speed until light and fluffy. Add molasses, egg and 1 teaspoon of the vanilla; mix well. Gradually beat in flour mixture on low speed until well mixed. Press dough into a thick flat disk. Wrap in plastic wrap. Refrigerate 4 hours or overnight.
Preheat oven to 350°F. Mix blackberry jam and rum extract in small bowl. Set aside.
Shape dough into 1-inch balls. Roll in granulated sugar to coat. Place 2 inches apart on ungreased baking sheets. Use a 1 teaspoon measuring spoon to make an indentation in center of each cookie. Fill indentations with blackberry jam mixture (about 1 teaspoon each).
Bake 8 to 10 minutes or until edges of cookies just begin to brown. Remove to wire racks; cool completely.
For the Chocolate Drizzle, place chocolate chips in microwave-safe bowl. Melt as directed on package. Stir in remaining 1 teaspoon vanilla. Transfer chocolate to large resealable plastic bag or piping bag. Drizzle over cooled cookies. Store cookies in airtight container up to 5 days.