Heat oven to 375°F. Grease a 9-inch square pan or a 10-inch ovenproof skillet.
Bloom yeast: In a bowl, stir warm water, warm milk, sugar, and yeast. Let sit until foamy, 5–10 minutes.
Mix the batter: Add melted butter, 1 1/4 tsp salt, and flour. Stir until a thick, sticky batter forms.
Spread batter into the prepared pan. Let rise while you cook the onions, 15–20 minutes.
Cook onions: Warm olive oil and 2 tbsp butter in a skillet over medium heat. Add onions, remaining salt, pepper, and caraway. Saute until soft and golden, 12–15 minutes.
Top batter evenly with the onions. Dust with paprika and scatter chives if using.
Bake until deeply golden and set in the center, 28–35 minutes. A toothpick should come out clean.
Cool 10 minutes before slicing. Serve warm.