Combine 1 cup flour, yeast, sugar, and salt in the bowl of a stand mixer fitted with the paddle attachment. Whisk to combine.
Combine water, milk and butter in a microwave safe measuring cup. Heat to between 120°-130°F (about 1 minute in my microwave). It’s best if you can use an instant read thermometer, but if you don't have one, heat your water to very hot bathwater temperature. Stir in potato flakes.
Pour the liquid ingredients into the dry ingredients and mix with the dough hook for 1 minute on medium speed. Add egg and 1 cup flour and mix another 1 minute. Add a third cup of flour and mix until your dough comes together to a soft dough. It will be slightly sticky to the touch. If needed, add up to ¼ cup more flour.
Turn out the dough onto a floured surface and dust it with flour. Knead until the dough comes together in a ball and the dough bounces back when pressed with two fingers. This will take about 4-6 minutes (longer if you've done this process by hand). Cover and let the dough rest for 10 minutes.
After 10 minutes, divide the dough into two pieces. Roll each into a 10-inch circle and cut into 12 wedges (like a pizza).
To form the crescent rolls, roll each triangle from the wide end to the point. Place rolls on a cookie sheet covered with parchment paper or a silicone baking mat, 1-inch apart. Cover with a kitchen towel and let sit at room temperature until doubled in size, about 1 hour.
Preheat oven to 375°F. Brush with melted butter. Bake for 15-25 minutes, or until light golden brown and cooked through. Brush with additional melted butter after baking.
Store rolls in an airtight container for up to 3 days. Baked rolls can be frozen for up to 3 months.
Make ahead instructions: Once you place the rolls on the cookie sheet you can cover them tightly with plastic wrap and store in the refrigerator up to 24 hours. Let come to room temperature and double in size before baking.