Heat oil in a skillet over medium-high heat and saute the chopped onion, season with salt to draw out the moisture. Cook for about 6-8 minutes until it changes color but not burn. Set aside and allow to cool down.
While the onion is cooling down, proof the yeast by combining yeast, water, and sprinkle a little bit of sugar, set aside and allow to foam up. It takes about 5-6 minutes. You can do the proofing in the mixer bowl if you're using a stand mixer or in a big bowl to do the whole mixing.
Add milk, butter, sugar, half of the sauteed onion, and egg to the yeast mixture. Combine the mixture on low speed with a dough hook, or just use a fork to mix if you're going to use your hand to mix the dough.
Whisk salt and flour then gently add to the yeast mixture and mix until it's smooth (you'll still notice the onion) and elastic. If you're using your hand transfer the dough to a floured surface and knead for a few seconds ( it doesn't require a very long kneading) and form a ball. Grease the mixing bowl and place the dough, cover, and place in a warm place and allow to double.
Place the doubled dough on a floured surface and punch down, knead to bring the dough together.
Grease a baking pan and set aside, divide into small balls or depending on how big you want it, tuck in the edges and place the tucked in part on the pan. Cover with a clean kitchen towel and allow it to double in size. It takes about 45 minutes -1 hour, at the final 15 minutes preheat the oven to 375 degrees F.
Brush the doubled dough with melted butter and sprinkle the rest of the sauteed onion over the dough.
Bake for 25 - 28 minutes or until golden brown.
Allow to cool down before separating them.
Serve and enjoy!