Vanilla Berry Ice Cream Cake

Ingredients

• 2 cups cake flour
• 2 tsp. baking powder
• ¼ tsp. salt
• 1 vanilla bean, split lengthwise
• 11 tbsp. unsalted butter, at room temperature
• 1 ⅓ cups sugar
• 3 large eggs plus 1 large egg yolk, at room temperature
• ¾ cup plus 2 tbsp. buttermilk, at room temperature
• 2 tsp. vanilla extract
• 1 cup blueberries
• 3 tbsp. sugar
• 1¼ tsp. cornstarch
• 2 tsp. cold water
• 2 tsp. freshly squeezed lemon juice
• 4 oz. strawberries, hulled, rinsed, and coarsely chopped
• 1½ tbsp. sugar
• 1 tsp. freshly squeezed lemon juice
• 2½ cups vanilla bean ice cream, slightly softened (homemade or store-bought)