Activate the Yeast: In a small bowl, combine the yeast, 1 teaspoon sugar, and ¼ cup lukewarm water (around 110°F). Let sit for 5–10 minutes until foamy and bubbly.
Make the Base: In a large mixing bowl, combine ¼ cup sugar, salt, and 2 tablespoons melted butter. Add 1 cup of warm milk or water. Let the mixture cool to lukewarm (this is important so you don’t kill the yeast).
Mix the Dough: Add the beaten egg and yeast mixture to the large bowl. Stir in 2 cups of flour until combined, then gradually add remaining flour, ½ cup at a time, until the dough forms a soft, smooth ball that’s no longer sticky.
Knead the Dough: Knead by hand for 6–8 minutes, or in a stand mixer with a dough hook for 5–6 minutes on medium speed, until elastic and smooth.
First Rise: Place dough in a greased bowl, cover with a damp towel, and let rise until doubled in size, about 1 to 1½ hours.
Shape the Rolls: Punch down the dough and divide into 9–12 equal pieces. Shape each piece into a smooth ball by folding the edges underneath. Place in a greased 9x9-inch baking pan.
Second Rise: Cover and let rise for 30–45 minutes, or until puffy. (For make-ahead rolls, cover tightly and refrigerate overnight. Let sit at room temp for 1 hour before baking.)
Bake: Preheat oven to 350°F. Bake rolls for 15–20 minutes or until golden brown and the centers reach 190°F.
Finish: Brush warm rolls with melted butter and serve.