Add the bacon to a medium pot over medium heat. Cook the bacon until the fat has rendered and the bacon is crispy, about 10 minutes. Remove the cooked bacon using a slotted spoon to a plate. Reserve as much of the fat in the pan as possible.

Remove all but 2 tablespoons (28g) of the bacon fat, add the shallots and cook until soft, about 8 minutes.

Add the sugar, tomatoes, apple cider vinegar, and salt, simmering over medium-low heat, cooking for about 15 minutes.

Add the bacon and continue cooking until a thick jam forms, about an additional 15 to 20 minutes. Remove from heat and let cool. Refrigerate in an airtight container until ready to use, up to 2 weeks.

Thinly spread one side of each bread slice with the mayonnaise. Heat a large skillet or cast iron pan over medium-low heat, add the butter and allow to melt.

Place half of the slices of bread, mayo-side down, on the pan or griddle. Divide the cheese evenly on top of the bread slices.

Top each with 3 tablespoons (55g) of the tomato bacon jam. Top each with a second slice of bread, mayonnaise side up.

When the underside is golden brown, about 3 minutes, use a spatula to flip the grilled cheese and continue to cook. Gently press down on the sandwich to encourage browning and continue to cook until the cheese has melted and the underside is golden brown, about an additional 3 minutes.

Remove to a cutting board and open each sandwich. Add the arugula, then close the sandwiches and cut each in half. Serve immediately.