Combine milk, sugar, butter, egg, and yeast in the bowl of an electric mixer with dough hook attachment. Let sit until mixture becomes bubbly or creamy, about 5 minutes. Add flour, cornmeal, and salt. Beat on low speed until flour is incorporated. Increase speed to medium and mix until dough is uniform and sticky, about 2 minutes.

Lightly dust work surface with flour. Transfer dough to work surface and gently shape into a ball. Place in a lightly oiled bowl. Cover and let rise in a warm place until nearly doubled, about one hour.

Combine the honey with lime zest and juice in a small bowl and set aside.

Turn out dough onto a lightly floured surface and roll to about 1/8 inch thick (you can work with half or quarter of the dough at a time, as desired). Using a knife, cut dough into 2 1/2- to 3-inch pieces (triangles, rectangles, squares, or circles -- any desired shape).

Fill a deep cast-iron pot with 4 to 6 inches of oil and heat to 375.

Working quickly and dropping only a few at a time, add dough to the hot oil. Using a spatula or a spoon, drizzle hot oil over tops ; this will cause them to puff. Flip sopapillas and cook until golden brown, about 20 to 30 seconds. Remove sopapillas to a platter or baking sheet lined with a kitchen towel to drain, and repeat with remaining dough. Sopapillas can be kept warm in a low oven (about 150 degrees) while frying remaining dough.

Serve hot, dusted with powdered sugar and drizzled or filled with the honey-lime mixture.