Combine the yeast and warm water in a large bowl, stirring briefly until well blended. Let this mixture sit for 2-3 minutes until you see bubbles forming on the surface, which means your yeast is active and ready.
Add the beaten eggs, melted butter, and brown sugar to the yeast mixture, stirring until everything is well combined and the sugar begins to dissolve.
Gradually add the flour and salt to the bowl, then use your hands to gently mix all ingredients until they form a cohesive dough. Don't knead at this stage - just mix until combined.
Cover the bowl with plastic wrap or a damp kitchen towel and place in a warm spot to rise for at least 1 hour, or until the dough doubles in size.
While the dough rises, prepare your coconut filling by combining coconut flakes, 2 tablespoons melted butter, brown sugar, and vanilla extract in a small saucepan over medium-low heat.
Simmer the filling mixture, stirring occasionally, until the liquid reduces and the mixture becomes thick and sticky. This usually takes 8-10 minutes. Remove from heat and let it cool completely.
Once the dough has doubled, punch it down in the center to release the air, then lightly dust with flour to prevent sticking.
Pinch off lime-sized portions of dough and flatten each piece by hand on a lightly floured surface to create small circles about 4 inches wide.
Place 2 tablespoons of the cooled coconut filling in the center of each flattened dough circle.
Gather all edges of the dough up and around the filling, pinching them together at the top and twisting to seal completely. Make sure there are no gaps where filling can leak out.
Place each filled dough ball seam-side down on a greased baking sheet, leaving about ½ inch space between each roll.
Cover the shaped rolls and let them rise in a warm place for another 30 minutes until they're puffy and slightly larger.
Preheat your oven to 350°F about 5 minutes before the second rise is complete.
Bake the rolls for 15 minutes or until the tops turn golden brown and sound hollow when lightly tapped.
Remove from oven and let the Pan De Coco rest on the baking sheet for 5 minutes before transferring to a cooling rack or serving immediately while warm.