Dissolve the yeast and sugar in the warm water in the bowl of a stand mixer fitted with a dough hook, and let stand until the liquid is foamy, about 10 minutes.

Crack two of the eggs and add them to the yeast mixture along with the oil. Whirl on low speed until blended. Scoop in the flour and salt and mix on medium-low speed, scraping down the bowl once or twice, until a smooth, slightly sticky ball that thwacks and cleans the side of the bowl forms, 7 to 10 minutes. Add up to 1/2 cup flour a bit at a time, if needed.

In a medium-sized bowl, add the shredded cheese and 2 tablespoons of the Taste of Portugal spice mix. Toss the cheese until combined with the spice. Set it aside.

Dump the dough onto a lightly floured surface and knead a few times until supple, adding more flour if needed. Flatten the dough into a large disk, sprinkle with the presunto, chouriço, and cheese mixture, and knead to distribute the pieces evenly. It will seem like a lot, but press on.

Place the dough in a large lightly oiled bowl, turn to coat, cover with plastic wrap, and let rise in a warm, draft-free spot until doubled in size, 1 to 1-1/2 hours.

Line a 13-by-18-inch rimmed baking sheet with parchment and sprinkle with cornmeal. Lightly dust a work surface with flour, turn the dough out, and knead several times.

Portion the dough into 2.25 ounce sized rolls and shape each piece into a ball. Transfer the balls to the baking sheet and let rise, covered with a tea towel, until doubled in size, about 1 hour.

Meanwhile, position a rack in the middle of the oven and crank up the heat to 425°F.

Whisk the remaining egg with 1 tablespoon of water and brush the rolls with the egg wash. Bake until the loaves are deeply brown and sound hollow when thumped on the bottom, 20 to 25 minutes. Transfer to a rack to cool completely. The loaves will keep for up to 2 days wrapped in plastic, but you can’t beat them on the day they’re made.