Beat the first 6 ingredients at low speed with an electric mixer for 1 minute. Scrape down sides and beat at medium speed 2 more minutes until well blended, scraping down sides if needed. Pour batter evenly into 3 greased and floured 9-inch round cake pans.
Bake at 350 for 20-23 minutes or until cakes are light brown and spring back when lightly pressed with your finger.
Cool in pans on wire racks for 10 minutes. Remove from pans; cool completely on wire racks.
Place on layer on a serving platter. Spread with 1/2 of blackberry jam, leaving a 1/2-inch border. Spread a smooth, thin layer of warm Quick Caramel Frosting over jam. Repeat layers. Top with third cake layer. Spread top and sides of cake with remaining Quick Caramel Frosting, working quickly before the frosting hardens. (If frosting becomes too hard to spread, place it over low heat for 1 minute, stirring constantly.) Let cake stand 10 minutes.
Quick Caramel Frosting:
Bring first 3 ingredients to boil in a 3 1/2-quart saucepan over medium heat, whisking constantly, about 2 minutes.
Stir in milk and bring to a boil. Remove from heat. Add powdered sugar and vanilla, stirring with a wooden spoon until smooth. Use immediately.