Make the starter: Use an active sourdough starter (this is ideal). Mix together the thick sweet starter. Leave it at room temperature to ferment and become active for 8 to 12 hours.
Warm milk: Measure out at least 170 grams of milk into a pot (some will evaporate while it is heated). Heat milk over medium heat until you begin to see steam rising off it or reaches a temperature of 115 – 120º F.
Scale: Add 150 grams of the warm milk to the mixing bowl, add in the sugar and 160 grams of the starter prepared in step 1. Allow the sugar to dissolve. (Remember to use the scale’s tare function to remove the weight of the bowl and previous ingredients to only weigh the current item added) Add the flour, lard*, and butter.
Mix: Mix ingredients until a shaggy dough is formed, then add vanilla extract, salt, 10 grams milk, and eggs. Continue mixing dough using the spoon and a dough scraper to scrape the sides of the bowl. (Mixing and kneading steps can be done in a stand mixer for 3 to 7 minutes with a dough hook on lowest two speeds, if available.)
Knead: Fold and knead dough 10-15 minutes or until dough is smooth, no longer sticky, and releases easily from the bowl or work surface.
Bulk fermentation: Cover the dough and allow to ferment until doubled in volume, about 6 to 10 hours. This dough is very rich, so give it time.
Prepare the baking sheets: Cut two large pieces of parchment. Place parchment papers on two flat baking sheets.
Make topping: Scale dry ingredients into a mixing bowl. Stir with a spoon to combine. Add lard, butter, and vanilla extract. Mix until a very smooth paste dough is formed. To make different colors or flavors, divide the paste dough. If dividing in half, for two flavors, add 2 grams of cinnamon powder or cocoa powder and mix until a cinnamon or cocoa colored paste dough is formed. If dividing into thirds for three flavors, add 1 gram of cinnamon or cocoa powder. Color can be added to the white vanilla paste dough as well.
Divide: Divide dough into 12 equally sized pieces (weighing about 87-88 grams each).
Shape: Gently press each piece into a small disc then fold edges into center, pinching together to form a seam. Turn dough over onto the seam and rotate dough on work surface with light pressure and quick movements until it becomes a smooth ball. Place ball of dough onto parchment paper lined baking sheet. Gently press down so that the roll is wider and stands about 1-inch high. Repeat with remaining pieces of dough. Balls of dough should be 2 to 3 inches apart on baking sheets.
Top: Lightly grease the top of the rolls. Divide the paste dough prepared in step 7. Scale 12 pieces weighing about 17 grams each. Roll a divided piece of the paste dough into a ball. Pass back and forth between the palms of your hands to flatten into a thin disc that is wide enough to cover the tops of the rolls. Press the disc on the top of the shaped rolls. Repeat until all the rolls are topped. If you have a concha press, use that to cut pattern into the paste dough. Otherwise, use a regular table knife to cut curved lines or a crosshatch pattern in the paste dough. Take care not to cut the dough of the roll underneath.
Proof: Cover rolls with plastic wrap and proof in a warm place for 2 to 4 hours, until at least 1.5 times larger, double in size is ideal.
Preheat: Preheat oven to 375º F.
Bake: Place rolls into the oven. Reduce oven temperature to 350º F. Bake rolls for 20 to 22 minutes until just turning light golden brown around the bottom.
Cool: Cool rolls on a cooling rack for at least 20 minutes before serving. These are best enjoyed fresh.