Stir together the soft butter, brown sugar, and cinnamon until it forms a thick, spreadable paste. Scoop into a piping bag or zip-top bag and snip a small corner. Set aside at room temperature - if it gets too firm, it won't pipe properly.

Beat the cream cheese until smooth, then whisk in the powdered sugar, milk, and vanilla until pourable but not runny. Add more milk by the teaspoon if needed. Set aside.

Whisk the flour, baking powder, sugar, and salt in a large bowl. In a separate bowl, whisk the milk, egg, melted butter, and vanilla. Pour the wet ingredients into the dry and stir just until combined - a few lumps are fine. Let rest for 5 minutes while you heat the pan.

Heat a nonstick skillet or griddle over medium-low heat and brush lightly with butter. Pour 1/4 cup batter per pancake. Immediately pipe a tight spiral of cinnamon filling starting from the center, leaving a 1/2-inch border. Cook until bubbles form on the surface and edges look set, about 2-3 minutes. Flip carefully and cook until the bottom is golden, 1-2 minutes more. Wipe the pan clean between batches - the sugar will caramelize and burn.

Stack the pancakes and drizzle generously with the cream cheese glaze. Serve immediately while the cinnamon swirl is still warm and gooey.