Preheat oven to 350°F. Line an 8-inch square pan with parchment, leaving overhang.
Whisk flour, baking powder, and salt in a medium bowl; set aside.
Stir melted butter, granulated sugar, and brown sugar in a large bowl until glossy.
Beat in the eggs and vanilla until smooth and slightly thickened.
Add dry ingredients in two additions, alternating with the milk. Mix just until combined.
Combine cinnamon and the 1/3 cup sugar in a small bowl. Stir in the 1 tbsp melted butter until sandy.
Spread half the batter in the pan. Sprinkle half the cinnamon mixture over it.
Dollop remaining batter on top and gently spread. Scatter the rest of the cinnamon mixture evenly.
Bake 23–26 minutes, until a toothpick comes out with a few moist crumbs.
Cool in the pan 20 minutes. Whisk powdered sugar, milk, and vanilla; drizzle over warm bars.
Let set, then lift out by parchment and cut into 16 squares.