Instruction

1. Heat milk and 3/4 cup water in a small saucepan over low until warm (110°F to 115°F). Transfer to a large bowl; add yeast and 1 tablespoon turbinado sugar. Let stand until surface bubbles form, 5 to 10 minutes. Add lard, salt, egg and egg yolk, 1 teaspoon vanilla, and remaining 1/4 cup plus 1 teaspoon turbinado sugar. Stir until combined. (It’s OK if some lumps of lard remain.) Add flour to bowl, 1/2 cup at a time, and bring dough together, first with a fork then with your hands, adding only just enough flour to create a sticky dough ball. (Flour your hands if dough sticks to them.)
2. Turn dough out onto a lightly floured surface. Knead, adding flour a little at a time as needed to prevent dough from sticking, until dough becomes cohesive and a bit shiny, about 8 to 10 minutes. Wash, dry, and grease bowl. Return dough to bowl. Loosely cover with plastic wrap or a clean towel; set in a warm, draft-free place to rise until doubled in size, 25 to 30 minutes. Meanwhile, melt butter in a small heavy-bottomed skillet over medium until it starts to brown, about 4 to 5 minutes. Remove from heat, and let cool slightly. Stir in remaining 1 teaspoon vanilla. Set aside.
3. Combine sugars, cinnamon, sea salt, and nutmeg in a medium bowl; set aside. Whisk together egg yolks and cream in a small bowl. Set aside.
4. Place dough on a lightly floured surface; divide into 3 even pieces, and shape each piece into a 20-inch-long rope. Roll and pat each rope into a 4- x 20-inch rectangle.
5. Generously brush dough rectangles with melted butter mixture, leaving a 3/4-inch border along long sides. Set aside 1/3 cup sugar mixture; sprinkle remaining sugar mixture over each dough rectangle, avoiding border. Brush one long edge of each rectangle with some of the egg wash. Roll up each rectangle, starting from the opposite long edge, to create a rope. Pinch seams to seal.
6. Press the 3 ropes together at one end. Tightly braid, and seal other end. Place braid on a large piece of parchment paper.
7. Connect the braid ends to form a circle- or oval-shaped ring; tuck ends under the ring. Transfer cake with parchment paper to a rimmed baking sheet; brush top with the remaining egg wash, and sprinkle with reserved 1/3 cup sugar mixture.
8. Set cake in a warm, draft-free spot. Cover loosely with a dish towel; let rise until doubled in size and dough is taut and springs back when lightly pressed, 25 to 35 minutes. Preheat oven to 375°F. Bake cake for 20 to 25 minutes, or until internal temperature registers 200°F. Let cake cool completely, about 1 hour and 30 minutes.
9. Cream together cream cheese, powdered sugar, and butter using an electric mixer, starting on low speed and gradually increasing to high speed, until smooth and fluffy. Reduce speed to low; gradually beat in cream. Add lemon juice, fine salt, and orange zest; beat on medium-low speed until smooth, about 5 minutes. Spread icing over top of cooled cake. Sprinkle icing with colored sugars.