Place an oven rack in the middle position and preheat your oven to 350°F (175°C). Use the softened butter to grease the inside of six 6-ounce ramekins, covering the bottom and sides.
Melt the 4 tablespoons of butter for the batter in the microwave in 20-second bursts or in a small saucepan over low heat. Set aside to cool slightly and let the milk sit on the counter to lose its chill. Fill your kettle and get it ready to boil.
In a medium mixing bowl, whisk together the flour, light brown sugar, baking powder, and fine salt for about 30 seconds until even in color. Press out any brown sugar lumps with the back of a spoon.
Pour the room temperature milk, cooled melted butter, and vanilla extract into the dry ingredients. Stir with a rubber spatula just until no dry flour remains. The batter will be thick and stiff, which is exactly what you want.
Divide the batter evenly among the six prepared ramekins, using about 1/4 cup (4 tablespoons) in each one. Use the back of a spoon or a damp fingertip to gently spread the batter into an even layer.
Bring the water to a full boil and measure out 1 1/4 cups. Add the dark brown sugar and 1 tablespoon of butter to a heat-safe liquid measuring cup, then carefully pour the boiling water over them. Stir for about 30 seconds until the sugar dissolves completely and the butter melts.
Hold a spoon upside down over one ramekin, with the back of the spoon just above the batter. Slowly pour about 1/4 cup of the hot topping over the back of the spoon so it lands gently on the batter. Repeat with the remaining ramekins, giving the topping a quick stir between each pour. Do not stir the ramekins after pouring.
Carefully slide the baking sheet into the center of the oven, keeping the ramekins level so the liquid does not slosh. Bake for 25 to 28 minutes, starting to check at the 25-minute mark. The cakes are done when the tops are golden, slightly cracked, and set to the touch, with thick sauce bubbling around the edges.
Use oven mitts to remove the baking sheet and move the ramekins to a cooling rack. Let the cakes rest for 5 minutes so the sauce thickens slightly. Sprinkle flaky sea salt over the tops, dividing it evenly among the six cakes.
Serve warm, right in the ramekins, with a spoonful of sweetened whipped cream or a scoop of vanilla ice cream melting over the top.