Add butter to a medium-sized pan over medium heat and cook until browned and fragrant, stirring the whole time, around 10-15 minutes. Remove from stove and pour into a heat-safe bowl and set aside to cool.
In a large mixing bowl, combine the eggs and sugar and beat at medium speed until the mixture becomes pale and thickens, around 5 minutes. Add the Eagle Brand® Sweetened Condensed Milk and vanilla bean paste and beat until incorporated.
Add the flour, salt, and baking powder and stir until just combined. Do not overmix. Pour the cooled browned butter into the bowl and gently fold thoroughly incorporated. Cover and chill for a minimum of 30 minutes.
Heat oven to 375 degrees and grease a madeleine tin with cooking spray. Remove chilled dough from the fridge and add one tablespoon of batter to each shell mold. Bake for 10 minutes, or until the edges are browned and the center is puffed. Remove madeleines from tin and cool on a wire rack.
To make the glaze, whisk together the powdered sugar, Eagle Brand® Sweetened Condensed Milk, and vanilla bean paste until a gooey ball forms. Slowly add water, one teaspoon at a time, until mixture reaches a drizzling consistency. Drizzle over madeleines, or dunk madeleines in the glaze, and set on a wire rack with a plate or bowl beneath to catch the drippings. Allow glaze to set before storing cookies, up to one week, in an air-tight container.