Heat a non-stick frying pan or skillet over medium-high heat. Add cooking oil. Add spinach to heated oil with a pinch of salt. Saute until spinach is wilted and most of the liquid has cooked off, 2 to 3 minutes.
Transfer spinach mixture to a mixing bowl and allow to cool slightly. Stir in mayonnaise and cheese.
Wipe out pan and return it to heat.
Assemble grilled cheese sandwiches by buttering one side of each piece of bread. Place half the slices of bread buttered-side down in hot pan. Top with spinach-cheese filling and then add remaining bread, buttered-side up. Cook until bottom of bread is golden brown, ~3 minutes. Flip and cook on the other side until filling is completely melted, ~3 minutes more.
While grilled cheeses are being made, heat up boxed soup in a saucepan over medium heat.
Slice grilled cheese in halves or into dippable strips and enjoy with your soup.