Pour ¼ cup melted butter into the bottom of a 5-quart slow cooker and tilt to coat the base and an inch up the sides.
In a medium bowl, whisk together the flour, 1 cup granulated sugar, 3 tablespoons cocoa, baking powder, ½ teaspoon salt, and espresso powder. Add the milk, ¼ cup melted butter, and vanilla. Stir just until smooth — about 30 seconds. Do not overmix. Spread the thick batter evenly over the butter in the crock.
In a small bowl, stir together the brown sugar, ⅓ cup granulated sugar, ¼ cup cocoa, and ¼ teaspoon salt. Sprinkle evenly over the batter. Do not stir.
Pour the boiling water gently and evenly over the surface, pouring over the back of a spoon. Do not stir.
Lay a clean kitchen towel or a double layer of paper towels across the top of the crock and press the lid on over it.
Cook on HIGH for 2 to 2½ hours or LOW for 3½ to 4 hours, until the top 1 to 1½ inches is set and springy and the edges pull slightly from the sides. Do not lift the lid during the first 90 minutes.
Turn off or switch to WARM. Rest uncovered 15 minutes — the sauce thickens as it sits. Scoop all the way to the bottom of the crock so each serving gets cake and sauce. Serve warm with vanilla ice cream.