Preheat the oven to 350°F. Grease and flour a 9x13 baking pan.
Beat the butter and sugar together until light and fluffy — about 5 minutes.
Beat in the eggs one at a time. Beat the batter for about two minutes after you add each egg.
In a medium bowl, mix together the flour, baking soda, and salt together. In a separate bowl, mix together the vanilla, mashed banana, and yogurt together.
Alternate between slowly mixing in the flour and banana mixture. Add the flour in three additions, the banana mixture should be added in two additions. Begin and end with the flour. Do not overmix.
Pour the batter into the baking pan and spread out. Smooth out the batter, but push the batter toward the edges of the pan so the middle of the pan has a small dent. This helps prevent the sheet cake from getting too puffy in the middle.
Bake for 30-35 minutes, or until a toothpick comes out clean.
Remove from the oven and allow to cool completely.
To make the frosting, beat the butter and the peanut butter together until a smooth consistency and one color. Add in the powdered sugar and vanilla and beat until smooth. Add in the heavy cream and beat on medium-high until light and fluffy. Once the cake is cooled, top with the peanut butter frosting.