Preheat oven to 350°F and spray three 8-inch round cake pans with nonstick spray. Line the bottom of the cake pans with parchment paper.
Whisk together flour, baking powder, baking soda, salt and cinnamon in a medium bowl.
Beat butter and sugars in a large mixer bowl at medium speed until light and fluffy. Beat in vanilla extract until incorporated. Add eggs one at a time and beat until combined. Reduce speed to low and add flour in 3 additions, alternating with buttermilk (begin and end with flour mixture).
Divide batter evenly between cake pans and smooth with an offset spatula. Tap pans sharply to reduce air bubbles. Bake for 22 to 24 minutes, until the cakes are golden, a toothpick inserted in the center comes out clean, and the edges are pulling slightly away from the pan.
Let cakes cool in pans for 15 minutes before inverting onto wire racks to cool completely.
Beat butter, cream cheese, and salt on medium speed until combined. Scrape down bowl and add brown sugar and vanilla extract; beat on low speed for 2 minutes. Add powdered sugar 1 cup at a time and beat well, scraping down bowl as needed. Chill frosting for 10 minutes before assembling the cake.
Place one cake layer on a rotating cake stand. Use an offset spatula to swirl frosting over the cake layer. Transfer 1 cup of frosting to a large piping bag and pipe a ring of frosting around the outer edge of the cake layer.
Use a clean offset spatula to spread about 3 tablespoons of peach jam over the frosting. Arrange peach slices in a circular pattern over the peach jam (within the frosting border). Add second cake layer and repeat with frosting, peach jam, and peach slices. Add top cake layer and chill the cake for 15 minutes.
Spread a thin layer of frosting on top and around the sides of the cake to create a crumb coat. Chill for 30 minutes, uncovered.
Add about 1 cup of frosting to the top of the cake, swirling generously over the edge. Use offset spatula to apply frosting to the sides of the cake while you rotate the cake stand. Chill frosted cake for 30 minutes before serving.