In a large bowl, stir together the warm milk and yeast. Let sit briefly until slightly foamy if using active dry yeast.
Add the sugar, flour, egg, salt, and softened butter to the bowl. Stir until a shaggy dough forms.
Transfer the dough to a clean work surface and knead for about 15 minutes using a push-pull motion with your palms. The dough will start sticky but will become smooth and elastic over time. Avoid adding extra flour.
Continue kneading until the dough passes the windowpane test — a small piece should stretch thin without tearing.
Place the dough into a lightly greased bowl, cover with plastic wrap or a towel, and let rise in a warm place until doubled in size, about 90 minutes.
Punch down the dough and divide it into 12 equal pieces.
Shape each piece into a smooth, tight ball by rolling gently between your palms or against the countertop.
Line a 9×13-inch baking pan or tray with parchment paper and arrange the rolls evenly spaced apart.
In a small bowl, whisk together the egg yolk and milk for the egg wash. Brush the rolls generously.
Cover loosely and allow the rolls to rise again for 30–45 minutes until puffy.
During the final 20 minutes of rising, preheat the oven to 350°F (175°C).
Brush the rolls with another layer of egg wash before baking.
Bake for 20–22 minutes or until deeply golden brown and shiny on top.
While the rolls bake, prepare the garlic butter by melting the butter in a saucepan over medium heat. Add the garlic and sauté for 1–2 minutes until fragrant. Stir in the parsley and remove from heat.
Brush the warm rolls generously with the garlic butter immediately after baking.
Serve warm and enjoy.