Classic Vanilla Birthday Cake With Caramel Pastry Cream

Ingredients

• ¾ cup unsalted butter, room temperature
• 1 ⅓ cup sugar
• 3 large eggs, room temperature
• 2 tsp vanilla extract
• 2 ½ cup cake and pastry flour
• 1 ½ tsp baking powder
• ½ tsp baking soda
• ¼ tsp salt
• 1 cup buttermilk, room temperature
• 3 tbsp water
• 1 cup granulated sugar
• 1 tbsp white corn syrup
• 1 cup whipping cream
• 1 recipe Basic Vanilla Pastry Cream, chilled
• 1 cup 2% milk
• ½ vanilla bean
• 1 ½ tsp vanilla bean paste
• 3 large egg yolks
• 3 tbsp sugar
• 2 tbsp cornstarch
• 2 tbsp unsalted butter
• 4-6 cups icing sugar, sifted
• 6 tbsp 2% milk
• 1 ½ tsp vanilla extract