Preheat the oven to 350°F (175°C). Butter a 2.5- to 3-quart baking pan.*
Place the cornstarch and 2 tablespoons of the milk in a saucepan, and stir until throughly combined. Add the remaining milk, sugar, vanilla bean paste, and salt. Heat over medium-low heat, stirring occasionally, until the mixture comes to a simmer.
Pour the sauce mixture into the prepared pan, and set aside while you make the cake.
To make the cake:
Whisk together the flour, sugar, baking powder, and salt. In a separate bowl, whisk together the milk, melted butter, egg, and vanilla.
Add the flour mixture to the milk mixture, and stir just until combined. Spoon or slowly pour the batter over the sauce mixture in the pan..
Bake for 35 to 40 minutes, or until a pick inserted into the center comes out clean. The top of the cake should be golden brown, and the sauce will be bubbling around the edges.
Cool the pan on a wire rack for about 15 minutes before serving. Serve warm.
Notes
*Make sure your pan is large enough to prevent bubbling over during baking. If you're worried about spills, place the pan on a rimmed baking sheet before placing in the oven.
Store leftovers in an airtight container in the refrigerator up to 4 days. Reheat before serving. The sauce will thicken and begin absorbing into the cake the longer it is stored.
To make individual servings, divide the two mixtures among eight buttered 8-ounce ramekins and place on a rimmed baking sheet. Reduce the baking time to 25 to 30 minutes.