In a small skillet or saucepan, melt the butter over medium heat. Let the butter simmer gently until it begins to brown, about 4 to 5 minutes. The butter will foam and then become clear, with the milk solids dropping to the bottom of the pan. Swirl the pan while the butter is cooking to make sure it browns evenly.
When the milk solids turn brown, remove from the heat. Spoon off about 3 tablespoons of the browned butter, making sure to add some of the browned bits on the bottom. Set the 3 tablespoons of butter aside to use for the sauce; the remaining butter will be added to the batter.
Preheat the oven to 350°F. Spray an 8-inch square or 11x7-inch baking dish with non-stick spray and set aside. In a small bowl, combine 3/4 cups of the brown sugar and the cornstarch. Mix together until there are no lumps and set aside.
Measure out 1/2 cup of the ale and set aside to be used in the batter. Pour the remaining ale into a measuring cup and add enough water to make 1 1/2 cups of liquid. Pour the liquid into a saucepan and place over medium high heat. Bring to a boil while making the batter.
In a mixing bowl, combine the flour, remaining 1/2 cup brown sugar, baking powder, and ½ tsp salt. Whisk together. Add the remaining browned butter to the mixing bowl along with the ale, beaten egg, and vanilla extract. Gently whisk the batter just until combined. Scrape the batter into the prepared pan and smooth the top.
Sprinkle the brown sugar and cornstarch mixture evenly over the batter - there will be a lot. When the liquid comes to a boil, remove from the heat. Add the reserved 3 tbsp browned butter along with the kosher salt. Carefully pour the hot liquid over the batter. Pour the liquid slowly over the back of a large spoon to slow the flow of the hot liquid and avoid making a crater in the batter.
After adding the hot liquid, carefully transfer the baking dish to the oven. Bake for 40 minutes or until the cake is cooked through. The cake is done when a toothpick or skewer inserted into the cake at an angle to avoid the sauce on the bottom comes out clean.
Allow the pudding cool for 10 to 15 minutes before serving. This pudding is best served warm on the day it was made. The sauce will thicken when the pudding cools and some will be absorbed into the cake, which is normal. Spoon the pudding into serving bowls and top with the warm sauce. Serve with vanilla ice cream if desired and enjoy!