In a large blender, blend basil leaves and water. Strain through a fine mesh sieve with a cheese cloth or coffee filter. Reserve basil discard for pesto.
In a large bowl, combine flour, basil juice, sugar, yeast, olive oil, and salt. Mix with a chopstick or fork until combined.
Cover bowl with a damp towel and leave in a warm place to proof for 2 hours.
In the meantime, combine 1 cup of basil and 1 cup vegetable oil the same blender. Strain through a fine-mesh strainer to make basil oil. Reserve basil scraps for pesto.
Coat a large baking dish or cast iron skillet with butter and a drizzle of olive oil to grease.
In your blender, add all basil discard, parmesan, nuts, garlic, salt, lemon juice, and olive oil. Blend until combined.
Once dough has proofed, transfer to your greased pan.
Proof in a warm place for another 1.5 hours until bubbly.
Once the dough is doubled in size, preheat oven to 425°F.
Drizzle the focaccia dough with your basil oil and dimple the dough with your fingers. Sprinkle on a couple teaspoons of sea salt and any leftover fresh basil.
Cook focaccia for 25 – 40 minutes, uncovered, until golden brown and crispy. The dough should spring back when touched and be medium gold-brown with green spots.
Cool and remove from pan. Slice into sandwich halves and slather with pesto. Top each slice with mozzarella and grill in a hot pan for 1-2 minutes, covered, until cheese is melted.
Remove from heat and add sliced green tomatoes. Top with your other focaccia half.