Warm the milk and water together until just warm to the touch, about 110°F. Pour into the bowl of a stand mixer and stir in the yeast and sugar. Let it sit for 5 minutes, until a little foamy.
Add the egg, softened butter, potato flakes, and salt, then add 3 1/4 cups flour. Mix with the dough hook on low until it comes together, then knead on medium-low for 6 to 7 minutes. The dough should be soft, smooth, and slightly tacky, pulling away from the sides but still sticking a bit at the bottom. If it's truly sticky, add more flour 1 tablespoon at a time. Resist adding too much; extra flour is what makes rolls dry.
Cover the bowl and let rise in a warm spot until doubled, 1 to 1 1/2 hours.
Pour 2 tablespoons melted butter into a 9x13-inch baking pan and brush it over the bottom and sides.
Punch down the dough and divide it into 16 equal balls (about 50 grams each if you're weighing). Flatten each ball into an oval about 4 inches long. Brush the top lightly with melted butter, then fold it in half lengthwise, not quite edge to edge, so the top half sits slightly back from the bottom. Press the fold gently to seal.
Arrange the rolls in the pan in 2 rows of 8, shingled so each one leans on the next with the folded edge facing up.
To make ahead, cover the pan tightly with plastic wrap and refrigerate for at least 8 hours and up to 24 hours. The slow, cold rise adds flavor and makes the timing easy.
On baking day, take the pan out of the fridge and let the rolls sit at room temperature, still covered, until puffy and nearly doubled, 45 minutes to 1 1/2 hours depending on how warm your kitchen is.
Heat the oven to 350°F. Bake for 25 minutes then pull out and brush tops with melted butter. Bake another 5 minutes or so, or until golden on top and 190°F in the center. Pull them as soon as they hit golden; a few extra minutes is the difference between tender and dry.
Brush the hot rolls generously with the more melted butter if desired and sprinkle with flaky salt. Serve warm.