For the sandwiches: Prepare a grill for medium-high heat.
Assemble 4 sandwiches by spreading 2 tablespoons fig jam on 2 of the multigrain slices and 2 of the brioche slices, then adding 2 slices of cheese to each and another slice of the same bread. Spread the butter on both sides of the assembled sandwiches.
Place the sandwiches carefully onto the grates of the grill and grill on each side until the bread has grill marks and the cheese is melted, about 5 minutes per side.
For the salad and dressing: Meanwhile, whisk together the olive oil, balsamic vinegar, balsamic glaze, honey, salt, pepper and lemon juice in a large bowl. Add the mixed greens, tomatoes and basil to the same bowl and toss to coat.
Serve each sandwich with a side of salad.