In the pot, bring the potatoes and water to a boil. Add 1/2 teaspoon of salt and allow to cook until fork tender. Remove the potatoes using a slotted spoon and add them to the work bowl of the stand mixer. Reserve the potato water.
Using the paddle attachment, mix the potatoes until the are smooth. Measure 1-1/2 cups of the reserved potato water, adding more water as necessary, and mix in with the potatoes. Allow to cool to about 75º F/24º C. The water should feel just barely warm.
In the medium bowl, stir together 1 cup of all-purpose flour with 1 teaspoon of yeast. Add the flour and yeast mixture to the potato mixture and mix well. The mixture will look very wet and look like a batter. Allow to sit for 5 minutes.
Add the softened butter and 1/2 tablespoon salt and mix well. Add the whole wheat flour and stir.
Add 1 cup of the all-purpose flour and mix. (So far you've used 2 cups of the all-purpose flour.) Remove the paddle attachment and attach the dough hook. Adding 1/2 cup of flour at a time, up to 2 more cups of flour, mix on low. The dough will still be very wet and sticky and will not come together in a ball like many other doughs. The dough hook will probably have dough dripping from it.
Pour the dough into a large, clean container for rising. Cover with plastic wrap or a lid and keep in a warm place until the dough doubles, about 2 hours. The surface will look bubbly and the dough will still be very sticky.
Shape and Second Rise
Generously flour your work surface. Pour the risen dough onto the work surface and knead it for several minutes, working in more flour in as you go. Use the bench scraper to clean the sticky dough off the work surface. The dough will be quite soft and loose.
To make a loaf, take half the dough and shape it into a log. Put into a medium loaf pan. To make rolls, take half the dough and divide into 12 pieces. Drop each piece into the muffin tin. To make focaccia, take half of the dough and shape into a loose ball. Put into an 8-inch round cake pan and brush the surface with olive oil, and sprinkle sea salt, chopped thyme, and rosemary on top. Use your fingers to dimple the surface. Cover and allow the dough to rise for about 30 minutes.
While the dough rises, preheat the oven to 450º F/232º C (425º F/218º C) if you are using dark nonstick pans). Brush the risen rolls with melted butter. Brush the risen bread with melted butter and slash on a diagonal in two places. Dimple the surface of the focaccia again, as needed.
Bake the rolls for about 20 minutes and the focaccia for about 25 minutes, and the loaves for about 35 to 40 minutes. The bread should be golden brown on top and feel a bit crusty to the touch. Remove from the pans and allow to cool completely on a wire rack, about 30 minutes, before slicing.