Step 1: Load the Pan and Start the Dough Cycle: Pour the warm milk, warm water, and 2 tablespoons of softened butter into the pan. Add the flour, dig a well and drop in the yeast, and place the sugar, egg, and salt substitute in a corner away from the yeast. Select the "Dough" cycle and press start. About 1.5 hours later the machine beeps, signaling knead and first rise are done.

Step 2: Divide and Roll the Dough: Turn the dough onto a lightly floured board and deflate it gently — press flat with your palm, then gather it back into a ball. Divide into 12 equal pieces (about 50g each). Roll each into a smooth ball: cup your hand over the piece and circle it on the board; surface tension pulls it smooth.

Step 3: Pan and Proof a Second Time: Place the 12 balls in a greased 9×13-inch pan (or an 8-inch round pan), leaving a little space between them. Cover with plastic wrap and set in a warm spot — about 28°C (82°F), such as an oven proof setting or a microwave with a cup of hot water — for 30–40 minutes, until the balls touch each other and have doubled.

Step 4: Egg Wash and Bake: When proofed, brush the tops with the reserved egg (or a little milk). Bake at 180°C (350°F) for 18–22 minutes, until the tops are golden and a tap sounds hollow. If the color lags in the last 3 minutes, switch on the broiler to finish.

Step 5: Butter and Cool: The instant they come out, brush the hot rolls with the remaining 1 tablespoon of melted unsalted butter — this locks in softness and adds a milky sheen. Let the pan rest on a rack for 5 minutes, then turn out. Pull apart or serve whole. They are softest while still warm.