To prepare the French onion soup, in a soup pot over medium-low heat, melt the butter and warm the olive oil. Add  the onions and thyme and stir to coat with the butter and oil. Increase the heat to medium and cook, stirring occasionally and keeping an eye on the onions so they do not scorch, until they start to brown, about 15 minutes. Reduce the heat to medium-low, cover the pan and continue cooking, stirring more often, until the onions are a rich brown color, about 40 minutes.

Stir in the sherry, increase the heat to high and cook, stirring constantly, until all of the sherry has evaporated, 2 to 3 minutes.

Stir in the stock and bring to a boil, then reduce the heat and simmer, partially covered, for 20 minutes. Stir in the salt and pepper. 

To prepare the panini, preheat a panini press according to the manufacturer's instructions. 

Brush one side of each bread slice with the olive oil. Lay the slices, oiled side down, on a clean work surface. Place the cheese, prosciutto and arugula on 4 of the bread slices. Top each with one of the remaining bread slices, oiled side up. 

Place the sandwiches on the preheated panini press. Close the lid and apply slight pressure. Cook without disturbing for 5 to 8 minutes. Open the lid and check for color and temperature; the cheese should be melted and the bread, golden. If the bread is sticking to the press, let it cook for a bit longer and it will unstick. If the press seems to be hotter on the bottom, flip the sandwiches halfway through to ensure even cooking.

Transfer the sandwiches to a cutting board and cut in half. 

If you don't have a panini press, you can grill the sandwiches in a hot fry pan: Brush the outside of the sandwiches with a little olive oil and place in the hot pan over medium-high heat. Press the sandwiches by placing another heavy pan on top and reduce the heat to medium-low. When the bottom is golden and crusty and the cheese has started to melt, flip the sandwiches and cook on the other side.

Serve the soup with the grilled panini, which can be dipped into the soup if desired.