In a large pot, melt 2 tbsp. butter over medium heat. Add the scallions and potato, season with salt and pepper and cook, stirring occasionally, until vegetables soften, about 5 minutes. Add the stock and 1 1/2 cups water; bring to a boil. Reduce the heat to medium and cook until the potatoes are tender, about 15 minutes.
Meanwhile, divide half the cheese among 3 slices of bread. Top with the sliced apple and the remaining cheese and bread. In a large skillet, melt the remaining 1 tbsp. butter with 1 tbsp. oil over medium heat. Add the sandwiches and cook until the bread is golden and the cheese melts, about 2 minutes per side. Cut each sandwich into 4 strips.
In a blender, working in batches if necessary, puree the soup with the spinach; season. Divide the soup among bowls; drizzle with the remaining 2 tbsp. oil. Serve each portion with 3 grilled-cheese segments.