Preheat grill or griddle while making the salad.
Mash garlic to a paste and transfer to a large bowl.
Whisk in lemon juice and 1/3 cup of the olive oil, and salt and pepper to taste.
Whisk in thyme if using.
Toss 3/4 of the dressing with the beans, cucumber, and tomatoes, and let sit to let flavors meld while you grill the tuna.
Season the tuna with soy sauce or tamari.
Brush tuna with olive oil on all sides.
Grill or pan-fry 1 minute on each side.
Remove from heat and set aside.
Taste beans and adjust seasoning if needed.
Cut grilled tuna into 1/3-inch slices.
Plate beans and lay tuna on top, napping with the remaining dressing and garnishing with additional thyme if desired.