In a small bowl, use a fork to thoroughly mix the softened butter, brown sugar, and cinnamon until it forms a smooth, spreadable paste. Transfer the mixture to a piping bag or a sturdy zip-top bag. Set aside at room temperature.

In a large bowl, whisk together the flour, sugar, baking powder, baking soda, and salt. In a separate medium bowl or large measuring jug, whisk the buttermilk, milk, eggs, melted butter, and vanilla until well combined. Pour the wet ingredients into the dry ingredients and gently whisk until just combined; a few small lumps are okay. Do not overmix. Let the batter rest for 5 minutes.

Heat a large non-stick skillet or griddle over medium-low heat. Lightly grease with butter or cooking spray. For each pancake, pour about 1/3 cup of batter onto the skillet. Immediately snip a small corner off your piping bag and swirl the cinnamon filling directly onto the wet batter in a tight spiral, starting from the center. Cook for 2-3 minutes, until bubbles form on the surface and the edges look set.

Carefully flip the pancake with a wide spatula. The side with the cinnamon swirl will now be face-down. Cook for another 1-2 minutes, until the pancake is golden brown and cooked through. Transfer to a plate. Repeat with remaining batter, greasing the skillet as needed.

While pancakes cook, make the icing. In a medium bowl using a hand mixer, beat the softened cream cheese and butter until smooth and creamy. Beat in the confectioners' sugar and vanilla. Add milk, one tablespoon at a time, until you reach a thick but drizzle-able consistency. Stack 2-3 pancakes per plate and generously drizzle with the cream cheese icing. Serve immediately.