Position a rack in the middle of the oven and preheat the oven to 350°F. Using nonstick cooking spray, lightly grease a 15x10-inch jelly roll pan.
In a small saucepan set over medium heat, melt the butter. Add the cocoa powder and whisk until there are no longer clumps, then add 1 cup of water. Bring to a rapid simmer, then remove from heat and set aside to cool slightly.
In a large mixing bowl, combine flour, brown sugar, baking soda, cinnamon, and salt. Whisk until well combined and there are no longer any clumps. Pour in the cocoa mixture and using a hand mixer, mix over medium-low speed until well combined, scraping down the sides as you go to make sure it’s well combined.
Add in 1/3 cup of the condensed milk, eggs, and vanilla. Beat again over medium-low speed until well combined.
Pour batter into the prepared pan and tap the pan on the counter to make sure it settles evenly across the pan. Transfer to the middle rack of the oven and bake until cake springs back lightly when touched, about 15 minutes. Remove from the oven and set aside to cool slightly while you prepare the icing.