Chop 1/4 of the basil leaves. Mix chopped basil, thyme, garlic, lemon zest, salt, and pepper in a bowl. Rub spice mixture on tilapia fillets and place in an airtight container. Marinate for 2 hours at room temperature.
Preheat the oven to 350 degrees F (175 degrees C). Line a baking dish with aluminum foil and grease with 1 tablespoon olive oil.
Place tilapia fillets into prepared baking dish, shaking off excess seasoning. Sprinkle cherry tomatoes, shallot, and remaining unchopped basil leaves on top. Drizzle 1/2 the lemon juice and 2 tablespoons balsamic vinegar over the fish.
Bake in the preheated oven for 10 minutes, until fish is starting to flake.
Meanwhile, combine remaining lemon juice, 1 tablespoon olive oil, and 1/4 cup balsamic vinegar in a saucepan and bring to a boil over medium-high heat. Boil mixture until it has the thick consistency of molasses, 5 to 10 minutes.
Set an oven rack about 6 inches from the heat source and preheat the oven's broiler.
Broil fish until tomatoes are charred, about 5 minutes. Remove from oven and discard liquid. Pour balsamic reduction on top of fish.