Mix the brown sugar and cinnamon in a small bowl. Set aside.
To make the cake:
Preheat the oven to 350°F (175°C). Grease a 9- x 13- x 2-inch baking pan. If you like, first line the pan with parchment paper.
Whisk together the flour, baking powder, baking soda, and salt. Set aside.
Using an electric mixer on medium speed, beat together the butter and granulated sugar until light and fluffy. Beat in the eggs one at a time, and then mix in the vanilla.
Reduce mixer speed to low. Add the dry ingredients to the butter mixture in three portions, alternating with two portions of the buttermilk. Mix just until combined.
Spread half of the batter evenly in the prepared pan. Sprinkle about two-thirds of the brown sugar-cinnamon mixture over the batter.
Carefully spread the remaining batter over the top. Try to cover all of the filling, but it's fine if you can't.
Sprinkle the remaining filling mixture over the top of the batter. Leave the topping as is, or use a thin knife to swirl it into just the top layer of batter.
Bake the cake for 35-40 minutes, or until a pick inserted into the center comes out clean.
Set the pan on a wire rack, and allow the cake to cool for 10-15 minutes.
To make the glaze:
Whisk together the confectioners’ sugar, 3 tablespoons (45ml) of the milk, and vanilla until smooth. Add more milk, a small amount at a time, until the glaze is thick but pourable.
Pour the glaze over the warm cake and spread evenly.
Allow the glaze to set before serving. Serve warm or at room temperature.
Notes
Store in an airtight container at room temperature up to 3 days.