Construct a charcoal fire or fire up the grill.

Add the thyme, garlic, mustard, olive oil, salt, and pepper to the ground beef in a big bowl. With care to avoid compressing the contents, mix the ingredients with a gentle fork. The beef should be formed into 12 equal patties, each measuring 2 inches.

To prevent the sliders from sticking, oil the grill grate when it is medium-hot. Sliders should be cooked for 4 minutes on the grill. With a spatula, flip the sliders over and cook for an additional 4 to 6 minutes, or cook them longer if you like your meat well-done. Place a half-ounce of Gruyere on top of each burger during the final two minutes of cooking, then cover the grill. Sliders should be removed to a dish and covered with foil.

The buns should be split in half crosswise, then toasted cut side down on the grill.

The 12 bottom buns should each have one slider on top, followed by a piece of tomato and red onion. Distribute the baby arugula among them. Serve hot with ketchup and cover with the top of the bun.