Preheat oven to 350 degrees F (175 degrees C). Grease 2 9-inch round cake pans, dust with flour, and line pan bottoms with waxed paper.
Whisk flour, baking powder, cinnamon, nutmeg, baking soda, and salt in a bowl until thoroughly combined. Stir buttermilk and vanilla extract in a separate small bowl.
Beat sugar and butter in a large mixing bowl, using an electric mixer on medium speed, until mixture is light yellow and fluffy. Scrape down the sides often while beating. Beat in egg and egg whites thoroughly, then beat 3/4 cup of blackberry jam into mixture.
Stir 1/3 cup flour mixture and 1/2 of the buttermilk mixture into moist ingredients, using a wooden spoon. Beat in 1/3 of flour mixture, remaining buttermilk mixture, and remaining flour mixture to make a batter. Spread batter evenly into prepared cake pans.
Bake in the preheated oven until a toothpick inserted into center of a cake comes out clean, 25 to 30 minutes. Let cakes cool in the pans for 10 minutes before removing, peeling off waxed paper, and letting cakes finish cooling on wire racks.
Place a cake layer, bottom side facing up, onto a cake plate. Spread remaining 3/4 cup of blackberry jam evenly over the cake; top with second cake layer with bottom side facing down.