These are very similar to your brioche bun recipe. You were right, they are maybe a tiny bit better. Maybe a little easier to make/less rising time. Instead of separating the dough into 9, I would do 8. Mine came out a little small.
1 tbsp. sugar
2 ¼ tsp. instant yeast
¼ cup warm water
1 cup warm milk
1 tbsp. vegetable oil
1 tsp. salt
3 to 3 ¾ cups all-purpose flour
1 egg beaten with 1 tbsp. cold water
sesame, poppy or caraway seeds, or coarse salt, for topping
In a bowl of an electric mixer, dissolve the sugar and then the yeast in the warm water. Add the milk, oil, salt and 1 ½ cups of flour to the yeast mixture. Beat vigorously for 2 minutes. Gradually add flour, 2 tablespoons at a time, until the dough begins to pull away from the sides of the bowl. Switch to the dough hook and knead until you have a smooth and elastic dough, about 7-9 minutes.
Transfer the dough to an oiled bowl. Turn once to coat the entire ball of dough in oil. Cover with a tightly-woven dampened towel and let rise until doubled, about one hour.
Turn the dough out onto a lightly oiled work surface. Working with oiled hands, divide the dough into 9 equal pieces. Shape each piece into a ball, and flatten into 3 ½ -inch disks. For soft-sided buns, place them on a well-seasoned baking sheet a half-inch apart so they will grow together when they rise. For crisper bun, place them three inches apart.
Cover with a towel and let rise until almost doubled, about 45 minutes.
15 minutes before you want to bake the buns, preheat the oven to 400°. Just before baking, brush the tops of the buns lightly with the egg wash and sprinkle with any desired toppings.
Bake for 15-20 minutes or until the internal temperature of the bread reaches 190°. When the buns are done, remove them from the baking sheet to cool on a wire rack. This will prevent the crust from becoming soggy.