For the Cake:
Preheat oven to 350 degrees . In a bowl, whisk the flour, baking powder and salt.
In an electric mixer fitted with the whisk attachment, beat the butter on high until creamy, 2 minutes. With the machine on, gradually add the sugar and beat, scraping the bowl, until fluffy, 8 minutes. Beat in the eggs one at a time, scraping the bowl. Beat in the vanilla. At low speed, add the flour mixture in three additions, alternating with the milk. At medium speed, mix until smooth. Transfer to prepared foil-lined 11-by-17-inch pan and bake for 30 minutes; invert and let cool on a rack.
For the Peanut Butter Frosting:
In an electric mixer, beat the butter and salt at medium speed until creamy about 1 minute. Add peanut butter and beat until creamy.
Sift in the confectioners' sugar, 1 cup at a time, mixing at low speed to incorporate (the mixture will be crumbly). Beat at medium speed for 1 minute, then increase to high speed and beat until smooth, 3 to 4 minutes. Gradually add the cream then beat until fluffy, 2 to 3 minutes.
Cut the cake crosswise in thirds, creating three 11-by-5 1/2-inch pieces. Stack the pieces, frosting side up. Spread with the remaining frosting. Pat the sides with 11/2 cups chopped salted peanuts. Cut crosswise into slices.