Preheat oven to 350 degrees. Butter and flour an 8-by-8-inch square pan. Sift together flour, baking powder, and salt; set aside.
In the bowl of a mixer fitted with the paddle attachment, beat butter and brown sugar on medium speed until creamy, about 3 minutes. Add eggs and vanilla, and beat until combined. With the mixer still on medium speed, add the flour mixture in two parts, alternating with the milk and beginning and ending with the flour; pour into pan.
Bake, rotating pan halfway through, until top is golden and springs back when touched, about 30 minutes. Let cool completely on a wire rack.
Assemble the cake: Cut cooled cake into two layers. Place the bottom layer on a wire rack set over a sheet pan, and pipe peanut butter filling on top. Carefully place remaining layer of cake over the filling. Pour half of the ganache over the top and gently tap pan against the countertop to spread evenly. Refrigerate until ganache sets, about 15 minutes. Repeat with remaining ganache. Chill until cake is firm, about 2 hours, and cut into 1 1/2-inch squares. Garnish with raspberry candies.