Serves: 32 rolls
Cook the potato in a microwave on HIGH until soft so it squeezes easily, using the designated baked potato setting or for up to 15 minutes. Scoop out the inside pass the warm potato through a potato ricer or food mill into a large bowl (you should have about 200g mashed potato).
Stir the hot water, eggs, half the sugar and the yeast into the potato. Add 280g of the flour and the salt and mix with a wooden spoon to make a sticky, shaggy dough. Cover the bowl tightly with cling film and set aside in a warm place until the dough doubles in volume, about 1 1/2 hours.
Beat the 110g of butter with the remaining sugar in a standing mixer fitted with the paddle attachment until light and fluffy. Switch to the dough hook and add the risen dough to the creamed butter. Continue to mix on low until the butter and dough come together, about 1 minute (stop the mixer and scrape down the bowl if needed; the dough will be very sticky). Gradually add the remaining flour, about 35g at a time, to make a shaggy dough that pulls away from the side of the bowl. Continue kneading on medium speed until the dough is smooth but still tacky, about 3 to 4 minutes.
Turn the dough onto a lightly floured work surface and knead by hand until it is smooth and no longer tacky, one to two minutes more. (If the dough is still sticky, gradually add 40g flour.)
Brush a large bowl with butter and turn the dough around in the bowl to coat lightly. Cover the bowl tightly with cling film, trace a circle the size of the dough ball on the plastic, and note the time. Leave to rise at room temperature until doubled in size, about one and a half hours.
Generously butter two 23-cm round or square cake tins. Turn the dough out of the bowl and pat into a rectangle about 35cm by 20cm, gently pressing out excess air. Divide the dough into 32 equal portions, using a pizza wheel. (Divide the dough in half lengthwise, then in half crossways. Cut each of those four sections into 8 equal-sized rolls.)
Tuck the edges of each piece of dough under to make round rolls and place seam-side down in the prepared tins, leaving a little space in between each roll (see Cook's Note). Cover the tin with buttered cling film and set aside in a warm place until the rolls rise almost to the rim of the tin and have more than doubled in size, about 45 minutes.
Position a rack in the centre of the oven and preheat the oven to 190°C / gas mark 5.
Bake the rolls until golden brown and puffy, and an instant read thermometer inserted into the centre of a roll registers 85°C, about 40 minutes. Remove the rolls from the oven and quickly brush the tops with soft butter.
Cool the rolls in the tin for about ten minutes before turning them out onto a rack in one piece. Cool slightly. Serve warm or at room temperature in one piece or pulled apart as individual rolls.
Cook's Note: At the point the rolls are formed, placed in the tin and covered, they can also be refrigerated overnight. If rolls have been refrigerated overnight, the final proofing time in step 7 will be closer to 1 1/2 hours.
Storing Tips: Storing and serving rolls in one piece keeps them moist and soft. Store baked and completely cooled rolls in one piece in an air-tight plastic bag, or wrapped loosely in cling film, then foil.