Prepare the brioche:

1. Activate the yeast: Dissolve the yeast in warm milk and let sit for 5 minutes.
2. Mix the dough: In a large bowl, combine flour, sugar, and salt. Add the eggs and yeast mixture.
3. Knead the dough: Gradually incorporate the softened butter and knead until the dough becomes smooth and elastic.
4. First rise: Cover and let the dough rise for 2 hours in a warm place.
5. Shape and proof: Shape the dough into a round disk, place it on a baking sheet, and let it rise again for 1 hour.
6. Bake: Preheat the oven to 350°F (180°C). Brush the brioche with egg wash, sprinkle with pearl sugar, and bake for 25 minutes. Let it cool completely.

Make the cream filling:

1. Heat the milk with vanilla until warm.
2. Prepare the custard: In a bowl, whisk together egg yolks, sugar, and cornstarch until smooth.
3. Combine: Slowly pour the hot milk into the egg mixture while whisking. Return to low heat, stirring continuously until thickened.
4. Cool and enhance: Remove from heat, stir in rum, and let cool.
5. Whip the cream: In a separate bowl, whip the heavy cream until stiff peaks form, then gently fold it into the custard.