In a medium bowl, combine black beans, cilantro, tomatoes, avocado, lime juice, and cayenne; stir to combine. Season with salt and pepper.
Rub steak with oregano; season with salt and pepper. In a 10-inch skillet, heat 1 teaspoon oil over high. Add steak to skillet, and cook, turning once, until browned and still rare, 2 to 3 minutes per side. (Steak will cook a little longer in the quesadillas during baking.)Transfer steak to a cutting board; let rest 5 minutes before slicing 1/4 inch thick.
Preheat oven to 425 degrees with racks in upper and lower thirds. Brush one side of each tortilla with remaining 2 teaspoons oil. Place 1 tortilla, oiled side down, on each of two baking sheets. Dividing evenly, layer with half the cheese, then onion and steak, ending with remaining cheese. Top with remaining tortillas, oiled side up. Press lightly to seal. Bake, turning over once (with a wide spatula), until lightly browned and crispy, about 5 minutes per side. Transfer to a cutting board; cut each quesadilla into 6 triangles. Serve with black-bean salsa and sour cream.