Preheat oven to 350°F (175°C). Place a 9-inch round cake pan or cast iron skillet in the oven with 6 tablespoons butter to melt and brown, about 5 minutes.
Remove the pan, and stir in the brown sugar and bourbon directly in the pan until well mixed. Spread evenly.
Toss sliced peaches with lemon juice and arrange them over the sugar mixture in a single, slightly overlapping layer.
In a bowl, cream softened butter and granulated sugar until fluffy (about 3 minutes).
Beat in the egg and vanilla extract until just combined.
In a separate bowl, whisk together flour, baking powder, and salt.
Alternately add dry ingredients and milk to the butter mixture, starting and ending with dry ingredients. Mix until just combined.
Spoon batter over the peaches and spread evenly without disturbing the fruit.
Bake for 40–45 minutes until a toothpick in the cake center comes out clean and the top is golden brown.
Let the cake rest for 5 minutes. Run a knife around the edges, place a serving plate on top, and flip. Let sit for 2–3 minutes before lifting the pan.