Melt your butter in a small saucepan. Add the non-dairy milk and water to the saucepan. Heat the liquid to between 105-115F (if using a thermometer) or until the water is warm to the touch but not piping hot or painful when you dip your pinkie finger in. (I usually melt the the butter, add the liquids to the saucepan, and heat for only 30 more seconds before the liquid is warm enough to use.) Set this mixture aside.
Whisk the flour, salt, sugar, and yeast together in a large mixing bowl. Pour in the wet ingredients. Mix with a spatula until you have a sticky dough ball.
At this point, you can knead the dough for about 5-6 minutes in a stand mixer fitted with the dough hook. (If you do this, you can also mix the dough in the stand mixer bowl, rather than a mixing bowl). If you don't have a stand mixer, lightly flour a surface and knead the dough by hand for 6-10 minutes. Either way, stop when the dough is very smooth and supple and withstands stretching.
Transfer the dough to a lightly oiled bowl, turning it once in the bowl to give it a coat. Cover the bowl with a cloth and allow the dough to rise for 1 hour, or until puffy and nearly doubled in size.
Preheat your oven to 350F. Grease a 9 x 5 x 2 1/2 inch or 8 x 4 1/2 x 2 1/2 inch loaf pan with vegan butter (or spray it with oil).
Lightly oil a countertop and turn the dough out onto it. Stretch it into a rectangle about 9 inches wide and 11 inches long. Have the wide side facing you. Starting at the top edge, roll the dough into a log. You don't want to roll too tightly, but you should create something compact, with a little surface tension. Gently tuck the side ends in and make sure the seam of the bread is underneath the roll. The finished roll should be about 4 by 8 inches.
Transfer the roll to the prepared loaf pan. Allow it to rise for another 35-60 minutes (this will depend on how warm your home is), until the dough has risen about 1 1/2 inches above the rim of the loaf pan.
Bake the loaf for 20 minutes. Rotate it in the oven to ensure even baking, then bake for another 15-20 minutes, or until it's light golden brown throughout and the bottom sounds slightly hollow when tapped (the center of the loaf should be at least 190F if you have a probe thermometer).
Remove the loaf from the pan and transfer it to a cooling rack. Allow it to cool completely before slicing and enjoying!