Drain the canned yams thoroughly, discarding the syrup. Weigh 180g of sweet potatoes, and mash or puree until completely smooth.
In the bowl of a stand mixer fitted with the dough hook, combine the flour, brown sugar, instant yeast, and salt.
Add the mashed sweet potatoes, milk, and egg. Mix on low speed until a shaggy dough forms. If dry flour remains after mixing, add water one tablespoon at a time, mixing thoroughly between additions. Only add enough to bring the dough together.
With the mixer running on low speed, add the softened butter a few pieces at a time. Continue mixing until the butter is fully incorporated.
Increase the mixer to medium-low speed and knead for approximately 5–8 minutes. The dough will be sticky and may stick to the sides of the bowl. It will firm up while proofing.
Wet your hands and gently transfer the dough to a lightly greased bowl. Cover with plastic wrap or a clean kitchen towel and let rise in a warm place for approximately 60–90 minutes, or until doubled in size.
Turn the dough onto a lightly floured surface and divide into 12 equal pieces, approximately 108g each. Shape each piece of dough into a smooth, round ball, using a little flour to keep it from sticking. Try to make the rolls uniform in size so they bake evenly. Watch the video in the recipe card to see how I shape mine! Grease a 10×10-inch baking pan, deep-dish pie plate, 10-inch cast iron skillet, or any ovenproof baking dish large enough to hold 12 rolls. Arrange the rolls evenly, leaving a little room for them to expand. Brush with melted or softened butter before proofing. Cover and let rise for approximately 45–60 minutes, or until the rolls are noticeably puffy and touching or nearly touching.
Preheat the oven to 350°F. Bake the rolls on the center rack for approximately 25–30 minutes, or until golden brown and the centers reach approximately 190°F. Allow extra time if needed for the centers to finish baking.
Remove the rolls from the oven and immediately brush the tops with melted butter. Allow to cool slightly before serving.