Combine the water and milk together in a microwave safe bowl. Microwave for about 20 seconds, until the liquid becomes just lukewarm. You should be able to insert your finger in the liquid and not feel a notable difference.
Add the brown sugar and sprinkle the yeast over the top. Stir. Set aside.
In a separate bowl combine 2 cups all-purpose flour and salt. Stir to combine.
The yeast should be frothy by now. Add 3 tablespoons of melted butter to the yeast and stir to combine. Create a well in the middle of the flour and pour the yeast mixture in the middle. Stir to incorporate. Use a spatula to sort of "knead" the dough in the bowl. Add up to 1/2 cup additional all-purpose flour as needed until the dough pulls slightly away from the sides of the bowl. It will still be sticky.*
Spray the dough ball in the bowl with vegetable spray. Cover with plastic wrap or a kitchen towel and place in a warm, dry place to allow the dough to rise until it's doubled in size, roughly an hour or more.
When the dough has risen, prepare a 9" square baking dish** by adding a tablespoon of vegetable oil. Spread the oil around the pan.
The dough will be sticky so sprinkle flour, over the top and gently work the flour into the dough. The dough should reach the point where it's still sticky, but will form into a ball.
Next, pinch off sections of the dough a little larger than a golf ball. Dip each ball in the oil from the pan and gently pull the sides down, working the dough until it's the shape of a ball. It can help you work the dough if you occasionally touch your fingers into the oil in the pan. Place the dough balls about 1" apart on your prepared pan.
Spray a sheet of plastic wrap with vegetable spray and place that over the rolls (so the plastic won't stick to the rising rolls). Cover and let rise 30-40 minutes until doubled in size.
When you're ready to bake, heat your oven to 350F. When the oven is at the correct temperature, remove the plastic wrap from the rolls, and place the pan in the oven. Bake for 25 minutes, until golden brown on top.
Remove them from the oven. Immediately coat with a thin layer of melted vegan butter. These rolls are best when served warm, but can be made ahead, frozen or refrigerated, and heated up prior to serving.
*Yeast rises best in a moist environment so lean on the side of the dough being too sticky rather than too dry.**If you want the rolls to rise nice and high, using a smaller dish (like a square 9X9 baking pan) is better.Make-ahead Tips: Cover the dough ball in plastic wrap and either freeze or refrigerate. In addition, you can freeze dough balls (before they have risen) by putting them on a tray and placing them in the freezer, once frozen, transfer to a freezer bag.If you're working with refrigerated or frozen dough, be sure to allow additional rising time in the subsequent step.To make evenly sized rolls, weigh the entire ball of dough and then divide by 12 and measure each individual dough ball.