Press tofu between paper towels to remove excess moisture, slice into 8 rectangles, toss in cornstarch, and pan-fry in 2 tablespoons olive oil over medium-high heat until golden on both sides (8 minutes total); meanwhile, halve cherry tomatoes, toss with 1 tablespoon olive oil on a sheet pan, and roast at 425°F (10 minutes).
Mince garlic and fresh basil leaves, whisk together with vegan mayonnaise to make herb aioli.
Spread herb aioli on both sides of all eight bread slices.
Layer two crispy tofu slices and one-quarter of roasted tomatoes on four bread slices; top with remaining bread slices to form sandwiches and serve immediately.