12 baba au rhum molds; an instant read or probe thermometer
Make the cakes: Heat the milk in a small saucepan over medium heat until it reaches 115 degrees F. Transfer the milk to the bowl of a stand mixer fitted with the dough hook. Add the yeast and 2 tablespoons of the brown sugar. Let sit until foamy, about 5 minutes.
Add the flour, the 5 tablespoons butter, the eggs, orange zest and 1/2 teaspoon salt and mix on low speed until combined, scraping the sides with a rubber spatula as needed, about 2 minutes. Increase the speed to medium-high and continue to mix until the dough is slightly large, elastic and everything is well combined (the dough will be very runny and soft), about 5 minutes. Cover the bowl with a clean dish towel and place in a warm spot to sit until doubled in size, about 1 hour.
Meanwhile, combine the currants and rye in a small bowl and let soak until the currants are plump, about 1 hour. Drain and set the currants aside.
Stir the apricot preserves with 2 tablespoons water in a small bowl until smooth; set aside.
Whisk the heavy cream with the remaining 2 tablespoons brown sugar in a medium bowl until stiff peaks form, about 5 minutes. Refrigerate until ready to serve.
Preheat the oven to 375 degrees F. Generously brush 12 baba au rhum molds with butter and place on a rimmed baking sheet.
Once the dough has risen, gently fold in the currants by hand. Scoop 1/4 dough into each mold. Cover the molds with a clean dish towel and place in a warm spot until doubled in size, about 20 minutes. Be sure to watch closely--the dough will rise quickly and you do not want it to rise above the mold.
Bake until the cakes are deep golden brown on top and have risen above the mold, 25 to 30 minutes. Let cool for 5 minutes on the baking sheet.
Meanwhile, make the syrup: Combine the granulated sugar, brown sugar, rye, rum and vanilla in a small saucepan and bring to a simmer over medium heat. Cook, whisking frequently, until the sugar has dissolved, 5 to 8 minutes. Remove from the heat and keep warm. Stir occasionally to prevent film from forming.
If needed, run a small offset spatula around the edge of each mold to loosen the cakes and pop them out. (The molds will still be warm so hold them with a clean dish towel to protect your hands). Working with one cake at a time, roll in the syrup to coat completely. Remove with a slotted spoon to a serving platter or plate and repeat with the remaining cakes. Spoon the apricot preserves over the cakes and then dollop each with some of the whipped cream. Serve immediately.