Homemade Mardi Gras King Cake with Cinnamon Filling

Ingredients

• 3/4 cup (180 ml) whole milk, warmed to 105°F-110°F
• 2 1/4 teaspoons instant yeast
• 1/3 cup (67 g) granulated sugar
• 6 tablespoons (85 g) unsalted butter, melted and cooled slightly
• 2 large eggs, room temperature
• 1 teaspoon vanilla extract
• 3 3/4 cups (450 g) all-purpose flour, plus more as needed
• 1 teaspoon fine salt
• 1/4 teaspoon ground nutmeg
• 1 teaspoon finely grated lemon zest
• 3 tablespoons (42 g) unsalted butter, softened
• 2/3 cup (142 g) packed light brown sugar
• 1 1/2 tablespoons ground cinnamon
• Pinch of fine salt
• 1 large egg beaten with 1 tablespoon milk, for egg wash
• 2 cups (240 g) powdered sugar
• 2-3 tablespoons milk
• 1/2 teaspoon vanilla extract
• Purple, green, and gold sanding sugar
• 1 plastic king cake baby, optional, inserted after baking