These soft, pull-apart rolls are similar in texture and flavour to a traditional Parker House roll, however, no dairy is used - making them perfect for any diets where dairy is restricted!
Add the potatoes to a medium saucepan and cover with cool water. Cover the pan, set over high heat and bring to a boil. Once boiling, reduce the heat to medium and simmer for 10 minutes - or until the potatoes offer no resistance when pricked with a fork. Drain and mash until smooth with a potato masher or fork. Set aside.
In a large mixing bowl (see note 3), whisk together the flour, salt and yeast. In a separate small bowl, whisk together the egg yolks, whole egg, olive oil and honey. Make a well in the flour mixture and add the mashed potatoes along with the egg mixture.
Use a wooden spoon or flexible spatula to mix until a shaggy dough forms. Turn out onto a clean work surface (resist the urge to add more flour unless the dough is unmanageably sticky) and knead until the dough is soft, smooth and lightly tacky - about 10 minutes (see note 4).
Gather the dough into a ball and set it in a clean mixing bowl. Cover with plastic (or a damp tea towel) and set in a warm place (preferably around 25°C/78°F) until doubled in size - about 45 minutes to 1 hour.
Once the dough has risen, turn it out onto a clean work surface (do not add any flour) and press lightly to de-gas. Portion the dough into 16 equal pieces (they will be about 50g (1¾oz). Working one piece of dough at a time, cup your hand over the dough and use it to drag it across the work surface to form a taut ball. Place each dough ball into a greased 20cm (8in) square baking dish in a 4x4 grid.
Cover the baking dish with plastic and allow to rise at warm room temperature until very puffed, the dough balls are all touching each other and they have doubled in size - about 1 hour (see note 5).
When the rolls are finishing their rise, preheat the oven to 180°C/350°F. Brush the dough with the beaten egg, sprinkle with flaky salt (if using) and bake until they are very puffed and deeply golden brown - about 25-30 minutes. Allow to cool in the pan for 5 minutes before unmoulding and serving.
Notes
Bread flour has a higher protein content than plain or all-purpose flour which makes gluten-development easier for these rolls. It's widely available at major supermarkets, however, if you don't have it, you can use plain/all-purpose flour in its place. Just note that you may need to knead the dough for longer before it reaches the desired stage of elasticity.
Use the weight measurement of salt over the volumetric if at all possible. The volume of different styles and brands of salt can vary drastically so going by weight will give you the most consistent results.
If you'd like to use a stand mixer, fit the mixer with a dough hook and follow step 2 as written. In step 3, mix the dough on low speed until a shaggy dough forms. Scrap down the sides and increase the mixer speed to medium-low and mix for about 5-7 minutes.
You will know the dough is ready when it passes the windowpane test. Simply pinch a ping pong ball-sized piece of dough and, using your fingers, gently stretch it as thin as possible. A thin sheet of dough should be able to form that allows light to pass through without tearing.
You can skip the second rise and move the rolls to proof in the refrigerator overnight. In the morning, let the rolls come to room temperature as the oven is preheating and follow the instructions in the final step.