Preheat oven to 350°F (175°C). Grease a 10-cup bundt pan with butter and dust with flour. Whisk flour, granulated sugar, baking powder, baking soda, and salt in a large bowl. In another bowl, whisk buttermilk, eggs, oil, and 2 tsp vanilla. Stir wet into dry until just combined.

Stir brown sugar and cinnamon in a small bowl. Spoon half the batter into the pan, spread evenly, and sprinkle all the cinnamon mixture on top. Add the rest of the batter, spread gently, then drag a knife through 4 to 5 times to swirl.

Bake 45 to 50 minutes, until a toothpick comes out with moist crumbs. Cool in the pan on a wire rack 10 minutes.

Beat cream cheese smooth. Add powdered sugar, 2 tbsp milk, and ½ tsp vanilla, whisking until thick and pourable. Add remaining milk a teaspoon at a time if needed.

Loosen cake edges with a knife, invert onto a plate, and drizzle glaze over the top. Serve warm.