Using a blender, mix 2 cups flour, 2 cups milk, 4 eggs, 1/2 teaspoon vanilla and 1/2 teaspoon salt until smooth; refrigerate.
In a medium saucepan, whisk together the cream, remaining 1 1/4 cups milk, 1/2 cup sugar and 1 pinch salt. Cook over medium heat until steaming, 2 minutes. In a medium bowl, whisk together the remaining 2 eggs and 1/2 cup sugar, then whisk in the remaining 2/3 cup flour until smooth. Gradually whisk half of the hot cream mixture into the flour-egg mixture, then whisk into the remaining cream mixture. Cook, whisking, over medium heat until thick, about 2 minutes; stir in the rum and the remaining 1 teaspoon vanilla. Scrape the pastry cream into a bowl and refrigerate, stirring occasionally, for 20 minutes.
In an 8-inch crepe pan or nonstick skillet, melt 1 teaspoon butter over medium-high heat. Add 1/3 cup chilled crepe batter and swirl to coat the pan. Cook until set on the bottom, about 1 minute; flip and cook for about 30 seconds more. Turn out onto a rack to cool. Repeat with the butter and remaining batter to make 10 crepes.
Place a crepe on a cake stand or flat plate; spread 1/3 cup pastry cream on top. Repeat with the remaining crepes and pastry cream, ending with a crepe. Refrigerate.
In a medium saucepan, whisk the chocolate, 3 tablespoons water, the corn syrup and remaining 2 teaspoons butter over medium-low heat until smooth. Let cool for 5 minutes, then spread on top of the cake, letting it drip down the sides. Refrigerate for at least 1 hour or up to 2 days before slicing.
The Do's and Don'ts of Cooking with Chocolate