PREHEAT OVEN TO 350°F AND GREASE A 9×13-INCH PAN WITH BUTTER OR NONSTICK SPRAY.
IN A LARGE BOWL, WHISK TOGETHER THE FLOUR, SUGAR, BAKING SODA, AND SALT. SET ASIDE.
IN A LARGE SAUCEPAN, HEAT THE BUTTER AND WATER UNTIL IT COMES TO A BOIL. REMOVE FROM HEAT AND WHISK IN PEANUT BUTTER AND OIL UNTIL SMOOTH.
LET COOL 2 MINUTES, THEN WHISK IN EGGS, MILK, AND VANILLA UNTIL WELL INCORPORATED.
ADD TO DRY INGREDIENTS AND MIX UNTIL SMOOTH. POUR BATTER INTO PREPARED PAN AND BAKE UNTIL A TOOTHPICK INSERTED INTO THE CENTER COMES OUT CLEAN, 40-45 MINUTES.
SET PAN ON A WIRE RACK TO COOL WHILE YOU MAKE THE FROSTING:
TO A MEDIUM SAUCEPAN, ADD THE BUTTER, PEANUT BUTTER, AND MILK AND BRING TO A ROLLING BOIL.
REMOVE FROM HEAT AND STIR IN VANILLA, FOLLOWED BY POWDERED SUGAR, 1 CUP AT A TIME.
WHISK UNTIL SMOOTH, REHEATING A BIT IF NEEDED, THEN POUR OVER CAKE. IF FROSTING IS STIFF, ADD A BIT MORE MILK.
WORKING QUICKLY, SPREAD FROSTING EVENLY OVER CAKE AND TOP WITH PEANUTS. SLICE, SERVE, AND ENJOY!