Place butter, sugar, and salt in the bowl of a stand mixer fitted with a dough hook. Pour the scalded milk into the bowl. Mix until the butter and sugar are dissolved, and the mixture cools slightly to 120°F or less, 6 minutes.
Add beaten eggs to the warm mixture and mix. Add 4 cups flour and the instant yeast. Use the stand mixer (or by hand) and knead the dough until it is smooth and beginning to pull away from the sides of the bowl, 6-8 minutes. If the dough is sticking to the sides of the bowl and is tacky, you will need to add more flour, 1-2 tbsp at a time, until smooth.
Once the dough clears the side of the mixing bowl and just slightly sticks to the bottom of the mixing bowl, remove the dough and knead a few times by hand on a lightly floured surface. The dough should be slightly tacky but not stick to the board while you are continually kneading.
Put the dough in a large oiled bowl to rise. Place a damp towel (or plastic wrap) over the bowl to ensure the dough will not dry out. Put the bowl in a warm place to rise until doubled, about 1 to 1 ½ hours.
Once risen, punch down the dough and cut it in thirds. Shape each third into a ball and punch out any air bubbles. Roll out each dough ball to a 15-inch circle (about ¼-inch thick) on a well-floured surface. Brush with melted butter, then cut into 12 triangles, as one would for a pizza. Start by cutting in quarters, then each quarter into three triangles. Take one triangle, stretch the end slightly, then roll up tightly. Pinch the ends to create a seal, then place the pinched end down on a parchment-lined baking sheet. Let the prepared butterhorns raise until they are about doubled in size (about 1 to 1 ½ hours).
Place into a preheated 350°F oven and bake for 8-15 minutes, switching racks and watching carefully to ensure the tops and bottoms do not overbake but turn lightly golden.
Remove from the oven and brush with melted butter. Let cool but serve warm.