Preheat oven to 350F. See above for a thorough list of tools and ingredients to get out before getting started.
Prepare the 10-in round or 9x9 square cake pan: Spray the pan with a generous coating of nonstick spray. Cut parchment paper to fit the bottom of your cake pan. It’s ok if it goes up the sides slightly. Press the cut parchment onto the sprayed pan. Set aside.
With the sifter over your large mixing bowl, sift the 1-1/2 cup of flour and 1 Tablespoon of baking powder. Mix in the generous pinch of salt. Set aside.
In a medium bowl, beat the 4 eggs, then whisk in the 1/2 cup of melted butter and 2 teaspoons of vanilla bean paste (or extract) until combined. Then add the 2-1/4 Tablespoons of ube extract and entire can of sweetened condensed milk and gently whisk til fully combined.
With the spatula, scrape the wet ingredients into the bowl of dry ingredients. Using a hand-mixer on low, beat just until combined, scraping the sides of the bowl, until no dry ingredients can be seen.
Pour the batter into the prepared pan and slide pan into the center of the preheated oven. Set timer for 40 minutes. (The more shallow the pan, the less time it will need. An 8-inch diameter may take 45-55 minutes, for example.)
Test the center of the cake with a toothpick or butterknife. It should come out with no batter clinging. For a 9x9 glass pan or 10-in round pan, it will probably take 40 minutes, but since ovens may vary, keep checking. It takes as long as it takes, and the center should be done before removing from the oven to cool.
Let cool completely before removing from pan. Then turn out onto a serving plate and right it. Ensure the cake is cool before applying any icing. If the parchment has stuck to the bottom of the cake, peel it off gently and discard.