Begin by heating a large skillet over medium-low heat and adding the butter until it melts and starts to foam, filling the kitchen with a rich aroma.
Add the thinly sliced onions into the skillet and sprinkle with salt. Stir gently to coat the onions in butter, then spread them out evenly in the pan.
Cook the onions slowly, stirring occasionally, for about 20-25 minutes. As they cook, they will soften, turn translucent, and begin to develop a golden hue, releasing a sweet aroma.
Once the onions are deeply golden and jammy, stir in the balsamic vinegar and cook for another 5 minutes, allowing the flavors to meld and the mixture to thicken slightly.
Remove the skillet from heat and let the caramelized onions cool for a few minutes, then transfer to a bowl.
Mix in the sour cream or Greek yogurt until well combined, creating a smooth and creamy consistency. Season to taste with freshly ground black pepper.
Transfer the dip to a serving bowl, smoothing the top with a spatula to make it inviting and even.
Serve the caramelized onion dip with chips, bread slices, or vegetable sticks for dipping, and enjoy the rich, sweet, and savory flavors.