In a large mixing bowl, combine warm milk, sugar, and yeast. Let stand for 5-10 minutes until foamy. Add melted butter, egg, and chopped thyme, whisking to combine.
Add flour and salt to the wet ingredients. Stir until a soft dough forms, then turn out onto a lightly floured surface and knead for 6-8 minutes until smooth and elastic.
Place dough in a greased bowl, cover with a clean kitchen towel, and let rise in a warm place for 1 hour until doubled in size.
Punch down dough and divide into 12 equal pieces. Shape each piece into a smooth ball by pulling the edges underneath. Place rolls in a greased 9x13-inch baking dish, spacing them evenly.
Cover rolls with a towel and let rise for 30-40 minutes until puffy and touching each other.
Preheat oven to 375°F. Brush rolls with melted butter and bake for 16-18 minutes until golden brown on top.
Remove from oven and brush with additional butter. Serve warm.