To make the crust:
In a medium bowl, combine the melted butter with the brown sugar, vanilla, and salt. Add the flour and mix just until blended. Don't worry if the dough seems too soft or oily; this is normal. Press the dough evenly and smoothly over the bottom of the lined pan. Bake for 15 to 20 minutes, until the crust is deep golden brown at the edges. Remove the pan from the oven and immediately sprinkle the chopped chocolate over the hot crust. Let stand for 2 to 3 minutes to melt the chocolate. Use a pastry brush or rubber spatula to spread the chocolate over the surface of the crust in a thin but thorough layer. Let the crust cool and then refrigerate it until the chocolate is set.
To make the filling:
In a heavy medium-large saucepan, whisk the sugar, cocoa, cornstarch, and salt to blend. Add about 3 Tbs. of the milk and whisk to form a smooth paste. Whisk in the remaining milk and the cream. Heat the mixture over medium heat, stirring constantly with a flat-ended heatproof spatula or wooden spoon and scraping the bottom, sides, and corners of the pan until the pudding thickens and begins to bubble at the edges, 5 or 6 minutes. Add the chocolate, vanilla, and rum and stir a bit faster to smooth out the pudding, cook the starch, and melt the chocolate, about 1 1/2 minutes. Scrape the hot pudding onto the crust and level it with one or two strokes of the spatula. Let the pudding cool, undisturbed (without mixing, jiggling, or spooning out a taste), at room temperature for 1 hour. Refrigerate the pan, uncovered, until the pudding is completely cool. Cover it and chill for at least several hours or overnight. Use the edges of the foil to lift the bars from the pan and transfer to a cutting board.
Cut into 20 bars or use round biscuit cutters for rounds, as pictured. Store leftovers in the refrigerator, as you would pudding.