Make the cinnamon swirl. Stir melted butter, brown sugar, cinnamon, flour, and a pinch of salt until smooth. Transfer to a squeeze bottle or a zip-top bag and set aside.
It should be thick but pipeable.
Whisk the dry ingredients. In a large bowl, combine flour, sugar, baking powder, baking soda, and salt.
Mix the wet ingredients. In a separate bowl, whisk buttermilk, egg, melted butter, and vanilla. Pour into the dry ingredients and stir until just combined. A few small lumps are fine.
If too thick, add a tablespoon or two of buttermilk.
Preheat the pan. Heat a nonstick skillet or griddle over medium to medium-low. Lightly butter the surface. Proper heat matters for clean swirls.
Pour and swirl. Pour 1/4 cup batter for each pancake. When bubbles form and edges look set (about 1–2 minutes), pipe the cinnamon mixture in a spiral from the center out.
Cook another 1–2 minutes until the underside is golden.
Flip carefully. Gently flip and cook 30–60 seconds more. If the swirl seems too melty, lower the heat slightly. Wipe the pan between batches if needed.
Make the glaze. Whisk powdered sugar, milk, vanilla, and a pinch of salt until smooth and pourable.
Adjust with more milk or sugar as needed.
Serve warm. Stack pancakes and drizzle with glaze. Add a pat of butter if you like extra richness.