Make the crepes: Combine the flour, eggs, milk, melted butter, granulated sugar, rum and salt in a blender; blend until smooth, scraping down the sides of the blender as needed. Set the batter aside, covered, 1 hour.
Line a baking sheet with parchment paper. Heat a 10-inch nonstick skillet over medium heat. Lightly brush with butter. Pour a scant 1/4 cup batter into the center, then quickly tilt and swirl the skillet to coat with the batter. Cook until the bottom of the crepe is golden brown and the top is set, about 1 minute. Flip and cook until lightly browned on the other side, about 30 more seconds. Invert onto the prepared baking sheet. Repeat with the remaining batter to make 16 to 20 crepes, stacking them as you go; lightly brush the skillet with more butter as needed. Don't worry if a few crepes tear-they can be layered in the middle of your cake.
Make the filling: Beat the butter and cream cheese in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Beat in the confectioners' sugar in 2 batches until fluffy, then beat in the rum.
Set aside the best crepe for the top layer. Put 1 crepe on a platter and top with about 1/4 cup filling, spreading it to the edge. Repeat, layering the remaining crepes and filling; top with the reserved crepe. Refrigerate until set, at least 2 hours or overnight.
Before serving, sprinkle the granulated sugar evenly over the top crepe; brulee with a kitchen torch. Chill 15 minutes before serving.