In a small bowl, combine the warm water (or potato cooking water), 1 teaspoon of the sugar, and yeast. Stir gently and let sit for 8 to 10 minutes until foamy.
In a large bowl, whisk together the gluten free flour, tapioca starch, remaining sugar, and salt.
In a separate bowl, combine the mashed potatoes, eggs, melted butter, and apple cider vinegar. Add the yeast mixture and stir until everything is smoothly blended. Pour the wet mixture into the dry ingredients and stir vigorously for about 2 to 3 minutes until a thick, smooth dough forms.
Grease a 9x5 inch loaf pan and scrape the dough into it. Smooth the top with a wet spatula. Cover loosely with greased plastic wrap and let rise in a warm spot for 55 to 70 minutes, until the dough has risen just above the rim of the pan.
Preheat the oven to 350 F. Bake the loaf for 38 to 45 minutes, until the top is golden brown and the internal temperature reaches 205 F. If the top browns too quickly, tent with foil during the last 15 minutes.
Let the bread cool in the pan for 10 minutes, then turn out onto a wire rack to cool completely. This bread is exceptionally soft and moist thanks to the mashed potatoes, making it perfect for sandwiches or toast.