Cook the finely diced yellow onion until lightly golden. Set aside to cool.
Making the Dough
Stir 2 teaspoon of sugar with the warm water (95°F to 110°F or 35°C to 43°C) and then stir in the yeast to dissolve. Set aside for the yeast mixture to foam.
Combine in the bowl of a stand mixture, the flour, the ¼ cup granulated sugar and onion soup mix.
Whisk the warm milk and room temperature egg, together in a separate bowl, until smooth. Melt the butter.
Add to the dry ingredients, the milk mixture, the yeast mixture, melted butter, and ⅔ of the cooked onions.
Use the dough hook attachment to mix the dough until it comes together. Transfer the dough to a floured surface. Fold the dough onto itself by kneading it for about 10 times. Shape into a ball of dough with a smooth surface.
Shaping and Baking the Rolls
Drizzle the mixing bowl with a little olive oil and place the ball of kneaded dough in the bowl. Cover with plastic wrap and let the dough rest for 20 minutes in a warm place.
Divide the dough into 12 equal pieces. Use a scale to find the total weight of the dough and then divide by 12 to get the weight of each bun. Eyeballing it also works by dividing in two and each half in two, until you have 12 pieces.
Shape each piece of dough over a non-floured surface. Shape your hand into a cup and place it over a ball of dough. With a rotating motion and a slight pressure against the dough, it should take the shape of a ball.
Place the rolled buns onto a parchment paper lined baking tray, in 3 rows of 4. Cover loosely with plastic wrap to keep them from drying and let them rise for 40 minutes or until doubled in size.
Prepare the egg wash by whisking together the egg yolk and 1 tablespoon of milk. Brush the tops and sides of the proofed buns. Divide the remaining cooked onions on top of each bun and sprinkle on the everything bagel seasoning.
Place the prepared Soft Onion Rolls in a preheated 375F oven. Bake for 30 minutes or until golden colored down the sides of the rolls.