Mix 4 large eggs, 4 cups plus 2 Tbsp. (516 g) all-purpose flour, ⅓ cup whole milk, 1 Tbsp. Diamond Crystal or 1¾ tsp. Morton kosher salt, 1 Tbsp. sugar, and one ¼-oz. envelope instant yeast (about 2¼ tsp.) in the bowl of a stand mixer fitted with the dough hook on medium-low speed until a shaggy, stiff dough forms around hook, about 10 minutes.

With the motor running, add 1 cup (2 sticks) unsalted butter, room temperature, cut into pieces, a piece at a time, mixing after each addition until almost completely incorporated before adding more, 10–14 minutes total.

Increase speed to medium and mix dough until soft, smooth, and elastic, 6–8 minutes.

Remove bowl from mixer and cover tightly with plastic wrap. Let dough sit at room temperature until puffy and almost doubled in size, about 1½ hours.

Coat pan with butter. Punch down dough; transfer to a surface. Pat and shape into about a 9"-diameter circle. Cut a 2"-long X in the middle, then transfer dough to pan, threading tube in center of pan through the X.

Press dough down into pan in an even layer and cover with plastic wrap; let sit at room temperature until puffy and it springs back when gently pressed, about 45 minutes.

Place a rack in middle of oven; preheat to 350°. Bake baba until top is deep golden brown, 40–45 minutes.

Transfer pan to a wire rack and let baba cool in pan 10 minutes. Carefully turn baba out onto rack and let cool completely, about 2 hours.

While the cake is cooling, combine 2 cups (400 g) sugar, 1 cup dark rum, 1 cup Cointreau or other orange liqueur, ⅔ cup fresh orange juice, ½ cup fresh lemon juice, and ¾ tsp. Diamond Crystal or ½ tsp. Morton kosher salt in a large, high-sided pot. Cover and bring to a simmer over medium heat.

Cook syrup just until sugar is dissolved, about 30 seconds; immediately remove pot from heat and let cool.

Poke 8 evenly spaced holes into top of baba with a chopstick, then poke many holes all over the bottom.

Pour half of syrup into reserved pan. Place baba back into pan and gradually pour remaining syrup over.

Let baba sit in pan, slightly pressing down on it occasionally to help it soak up the syrup, at least 1 hour and up to 2 hours.

Place a large rimmed plate upside down over pan and in one rapid motion turn over; slide away pan, allowing any excess syrup to drip down sides of baba and onto plate.

Briskly whisk 1 cup heavy cream in a large bowl until medium-stiff peaks form, about 2 minutes.

Transfer to pastry bag fitted with star tip and pipe 8 mounds over holes on top of baba. Top mounds of whipped cream with 8 maraschino cherries. Serve baba in slices with extra whipped cream on the side.