Make the doughnuts: Preheat the oven to 350 degrees and grease 2 nonstick 6-count doughnut pans with cooking spray. In a large bowl, whisk together the flour, baking powder and salt.
In a medium bowl, whisk together the eggs, melted butter, brown sugar, dulce de leche, milk and vanilla until smooth; add to the flour and whisk until combined. Spoon into the prepared doughnut pans until ¾ full and bake until a toothpick inserted comes out clean, about 16 minutes; let cool slightly.
Make the glaze: In a medium saucepan, melt the brown sugar with the butter and salt over moderate heat, stirring occasionally. Bring to a boil and stir in the dulce de leche, milk, corn syrup and vanilla. Boil for 3 minutes, stirring occasionally. Add the confectioners’ sugar and stir until combined.