To a mixing bowl, add the bread flour, salt, and instant yeast, and stir together. Add the non-dairy milk, the mashed potato, and the olive oil. Stir to combine until a dough forms. It will generally seem dry at first and then come together.
Transfer the dough to a lightly floured surface and knead for 5 minutes, or until elastic. (Alternately, use a mixer with a dough hook). Add as little extra flour as you can get away with during this process.
Once done kneading, place the dough in a bowl and cover it with a damp towel. Place it in a warm location inside your home, such as near the oven. In winter I sometimes turn on my oven for about 60 seconds to get some heat circulating, then turn it back off and put the bread inside. Let the dough rest until approximately doubled in size, which is generally about 30 to 45 minutes using the rapid-rise yeast, depending on ambient conditions.
Line a baking sheet with parchment paper. Divide the dough in half and shape each half into an oblong ball, and place on the baking sheet. If desired, use a sharp knife to score the top of the loaves. Let the shaped loaves rest for another 10 minutes while preheating the oven to 425 degrees Fahrenheit. Don't worry if they look a little flat; this bread gets a lot of oven spring.
Bake the vegan potato bread for approximately 40 minutes, or until the crust is browned all over the outside, and when tapped the bread sounds hollow inside. Remove from the oven, and let cool for at least an hour before slicing. If you let it cool for a few hours, you'll have the best results. You can slice it earlier than that, but it may still be a bit moist (as in my photos) and not as good for spreading things onto it. If you let it cool completely, the potato bread will have a nice tight crumb.
POTATO COOKING: Boiled potatoes work best since the potatoes supply a lot of the moisture for the bread. You can use roasted or steamed potatoes, and may just need to add some extra liquid to bring the dough together - totally fine. Leftover mashed potatoes work too!