Mix the Wet Ingredients: Use a stand mixer fitted with the paddle attachment to combine the eggs, softened butter, 1 cup sugar, and kosher salt. If you do not have a stand mixer, cream the softened butter with a fork, then whisk in the sugar, salt, and eggs until combined. Warm the Milk: Heat the whole milk in a small saucepan until it reaches 115°F. Pour half of the warm milk into the egg and butter mixture. Transfer the remaining half to a cup or small bowl.
Proof the Yeast: Stir 1 teaspoon sugar into the reserved warm milk, then add the active dry yeast. Mix gently and let it stand for 10 minutes, until the yeast becomes foamy and increases in volume.
Form the Dough: Add the proofed yeast mixture to the bowl containing the wet ingredients. Replace the paddle attachment with the dough hook. Mix on low speed while adding the flour ½ cup at a time. Continue until the ingredients come together and a ball of dough forms around the dough hook. Knead the Dough: Turn off the mixer and transfer the dough to a floured surface. Knead it by hand for a full 7 minutes, working the dough until it is properly kneaded and cohesive.
First Rise: Shape the dough into a ball and place it in a greased bowl. Cover the bowl with plastic wrap and let the dough rise in a warm place for 1½ hours, or until it is almost doubled in volume.
Divide the Dough: Remove the risen dough from the warm oven or rising area. Remove the plastic wrap and punch the dough down to release the trapped air. Divide the dough in half for two loaves, shaping each portion into a ball, or divide it into 16 portions for individual rolls. Shape the Rolls: If making rolls, shape each portion into a tight ball by cupping your hands around the dough and turning it against a flat surface.
Arrange the Dough: Place the shaped rolls in a greased 10-inch round pan, or use two greased 8-inch round pans when making two loaves. Cover the pan or pans with a kitchen towel.
Second Rise: Let the shaped dough rise in a warm oven for 1 hour. The dough should expand noticeably during this second rise.If using the oven as a warm rising space, do not heat the oven while the dough is rising. Prepare the Oven: After the second rise, preheat the oven to 375°F. Do not preheat the oven while the dough is still rising inside it.
Make the Egg Wash: Whisk 1 large egg together with 2 teaspoons water until combined. Gently brush the egg wash over the surface of the shaped bread using a pastry brush.
Bake the Bread: Bake at 375°F for 25 minutes. Begin checking the color around the 15–20 minute mark. If the bread is becoming too dark, carefully cover it with a sheet of foil and continue baking for the remaining 10–15 minutes.
Cool the Bread: Allow the baked bread to cool completely in the pan for 30–40 minutes before removing it.
Remove and Slice: Turn the pan over to release the loaf or rolls. Slice the loaves with a serrated knife. For rolls, separate them using a knife or gently pull them apart by hand.
Serve Warm: Serve the Portuguese sweet bread warm with butter.
