Step 1: Cook egg noodles according to package directions until al dente. Drain well, rinse with cold water, and toss with a tablespoon of olive oil to prevent sticking. Set aside.
Step 2: In a large, heavy-bottomed skillet or Dutch oven, melt 2 tablespoons of butter and 1 tablespoon of olive oil over medium-low heat. Add the thinly sliced onions, a pinch of salt, and a sprinkle of sugar. Cook, stirring occasionally, for 30-45 minutes, or until deeply golden brown and very soft and jammy. Remove from heat and set aside.
Step 3: While onions caramelize, prepare the brown butter. In a small, light-colored saucepan, melt 1/2 cup unsalted butter over medium heat. It will melt, then foam. Continue to cook, swirling occasionally, until golden-brown bits appear at the bottom and the butter smells nutty. Immediately pour into a heatproof bowl to cool slightly.
Step 4: Preheat oven to 375°F (190°C). Grease a 9×13-inch baking dish.
Step 5: In a large mixing bowl, whisk together the eggs, sour cream, milk, salt, and black pepper. Stir in the slightly cooled brown butter.
Step 6: Add the cooked and drained noodles and the caramelized onions to the egg mixture. Toss gently to ensure all ingredients are evenly coated.
Step 7: Pour the noodle mixture into the prepared baking dish, spreading it evenly.
Step 8: Bake for 45-55 minutes, or until the kugel is golden brown on top and set in the center. A knife inserted into the center should come out mostly clean.
Step 9: Let the Noodle Kugel With Caramelized Onions & Brown Butter rest for 10-15 minutes before slicing and serving. Enjoy!