Preheat oven to 350°F (175°C). Grease a 9-inch round springform pan.
Whisk together the flour, baking powder, cinnamon, and salt. Set aside.
Using an electric mixer on medium speed, beat the butter and sugar until light and fluffy. Add the egg and vanilla, and mix well.
Reduce mixer speed to low. Add the flour mixture in three portions, alternating with two portions of milk. Mix just until combined. (The batter will be thick.)
Spread 3/4 of the batter in the prepared pan. Set aside the remaining batter.
To make the filling:
Using an electric mixer on medium speed, beat the cream cheese, sugar, flour, and vanilla until combined. Add the eggs, and beat on low speed just until combined.
Pour the filling mixture over the cake batter in the pan.
To make the cinnamon swirl and finish assembling the cake:
Combine the brown sugar, butter, and cinnamon. Drop by tablespoonfuls over the filling.
Use a thin knife to swirl the cinnamon mixture into the filling.
Drop the remaining cake batter by tablespoonfuls over the filling.
Place the pan on a lined, rimmed baking sheet to catch any potential spills or leaks.
Bake 70 to 80 minutes, or until the center is almost set. If necessary, cover the top loosely with foil during the last 20 to 30 minutes to help prevent overbrowning.
Cool the cake in the pan on a wire rack for 10 minutes. Then, carefully run a thin knife around the edges of the pan. Cool for 1 hour. Then remove the sides of the pan.
Notes
This cake is best served warm. Refrigerate any leftovers and reheat servings as needed.