Position racks in the upper and lower thirds of the oven and preheat to 375 degrees. Line 2 rimmed baking sheets with parchment paper. On a cutting board, brush the tortillas with 2 tablespoons olive oil, then stack and cut into quarters. Arrange on the baking sheets in a single layer; season with salt and pepper. Bake until crisp and golden, 8 to 10 minutes.
Meanwhile, in a large skillet, heat the remaining 1 tablespoon olive oil over medium-high heat. Add the onion and garlic and cook until lightly browned, about 4 minutes. Add the broccoli and 1/3 cup water and cook, stirring, until the water is cooked off and the broccoli is tender, about 4 minutes. Stir in the beans and cook until heated through, 1 to 2 minutes; season with salt and pepper.
Sprinkle 1 cup cheese over each tray of chips; spoon the broccoli mixture on top, sprinkle with the remaining cheese and bake until melted, 1 to 2 minutes. Serve with the salsa and yogurt.