Melt the butter in a small skillet over medium heat. Add the onions and 1/2 teaspoon of the salt and cook until softened and caramelized, 18 to 20 minutes. Transfer to a small bowl and cool completely.
Pour the water into the bowl of a stand mixer with the hook attachment and sprinkle in the yeast. Let the mixture rest until it bubbles, 4 to 6 minutes.
Meanwhile, sift together the flour, sugar and remaining 1/2 teaspoon salt into a large bowl and stir to combine. Add the flour mixture and caramelized onions into the mixer and mix on the lowest speed until the dough comes together. Increase the speed to medium and mix until the dough is elastic and smooth, 6 to 8 minutes.
Grease a large mixing bowl with 1 teaspoon of the oil. Form the dough into a ball and place it in the bowl, gently turning until coated. Cover with a clean, damp towel and let rest in a warm place until the dough rises 1 1/2 times in size, 35 to 40 minutes. Meanwhile, line a baking sheet with parchment paper and evenly coat it with the remaining oil. Set aside.
Transfer the dough to a clean, lightly-floured surface. Punch down the dough and knead until smooth and no longer sticky, 6 to 8 minutes. The dough should spring back when you lightly press it with a finger. Divide the dough into 8 equal pieces, about 4 ounces each, and form into 4-inch-long rolls with slightly rounded, tapered ends. Place the rolls on the prepared baking sheet, making sure there is enough room between each to allow it to double in size. Cut three 1/4-inch-deep diagonal slashes across the top of each roll, then cover with a damp towel and set in a warm place to rise until the rolls almost double in volume, 20 to 25 minutes.
Place one rack on the very top shelf of the oven and another on the bottom shelf. Preheat the oven to 425 degrees F. Stir together the pretzel salt and caraway seeds in a small bowl and set aside.
Line 1 baking sheet with a dry towel and another baking sheet with greased parchment or a silicone mat. Bring 8 cups of water to a boil in a deep straight-sided skillet, and then stir in the baking soda. Boil 4 of the rolls, flipping in the water, 1 minute, and then transfer to the towel-lined baking sheet using a slotted spoon. Repeat with the remaining rolls. Let the rolls dry for 1 minute before transferring to the parchment-lined baking sheet and sprinkling with the caraway salt.
Bake the rolls on the top shelf of the oven until deep golden brown, 10 to 12 minutes, and then switch to the bottom shelf and continue baking until cooked through, 2 minutes longer.
Let the rolls rest 10 minutes on the baking sheet. Serve warm, with good German mustard or use as rolls for various sandwiches.
For miniature buns, divide the dough into 16 pieces and roll each into a ball. Reduce the baking time for the miniature buns to 8 minutes on the top rack of the oven.
Recipe courtesy of Susan Vu for Cooking Channel