Adjust oven rack to middle position and heat oven to 350 degrees. Generously grease two 8-inch square baking pans and cover pan bottoms with squares of parchment paper. Grease and flour parchment. Mix milk, egg whites, and vanilla together in 2-cup measuring cup.
Mix flour, sugar, baking powder, and salt together in bowl of electric mixer at low speed. Add butter and beat at low speed until mixture resembles moist crumbs, with no powdery streaks remaining.
Add all but 1/2 cup of milk mixture to crumbs and beat at medium speed (or high speed if using handheld mixer) for 1 1/2 minutes. Add remaining 1/2 cup of milk mixture and beat 30 seconds more. Stop mixer and scrape sides of bowl. Return mixer to medium (or high) speed and beat 20 seconds longer.
Divide batter evenly between three prepared cake pans and using rubber spatula, spread into even layer. Arrange pans at least 3 inches from the oven walls and 3 inches apart. (If oven is small, place pans on upper middle and lower middle racks in staggered fashion to allow for air circulation.) Bake until thin skewer or toothpick inserted in the center comes out clean, 21 to 25 minutes.
Rest cakes in pans 10 minutes, then loosen from sides of pans with a knife. Invert onto large plate, reinvert onto wire racks. Cool completely, about 1 1/2 hours.
**If you’ve forgotten to bring the milk and egg white mixture to room temperature, set the bottom of the measuring cup containing it in a sink of hot water and stir until the mixture feels cool rather than cold, around 65 degrees. Cake layers can be wrapped and stored for one day.
FOR THE FROSTING: Using stand mixer fitted with whisk, whip butter, peanut butter, cream, and vanilla on medium-high speed until combined, about 30 seconds. reduce speed to medium-low and slowly add sugar until smooth, 1 to 2 minutes. increase speed to medium-high and whip frosting until light and fluffy, about 5 minutes.
TO ASSEMBLE: Slice each cake in half horizontally; spread 1/2 cup jam over each bottom half and replace top half. Place 1 reassembled cake on cake plate, then spread 1 cup frosting in even layer over top. Place second reassembled cake on top of frosting. Spread remaining frosting over top and sides of cake. Press peanuts onto sides of cake. heat remaining 1/4 cup jam in bowl in microwave until melted and smooth, about 20 seconds. Transfer to piping bag or small plastic sandwich bag with 1 corner cut off. Pipe jam in straight lines over top of cake and lightly drag paring knife through lines to create marbled appearance. Serve.