Press and Slice the Tofu: Place the block of tofu in a tofu press for at least 15 minutes to remove excess water. Alternatively, wrap the tofu in several layers of paper towels, place it on a plate, and set heavy books on top. Once pressed, slice the tofu horizontally into eight 41-inch thick medallions.
Marinate the Tofu: In a shallow dish, whisk together 1 tbsp olive oil, lemon juice, minced garlic, oregano, and 81 tsp salt. Gently place the tofu medallions in the marinade, turning to coat each side. Let the tofu marinate at room temperature for at least 30 minutes.
Prepare the Vinaigrette: While the tofu marinates, combine all vinaigrette ingredients (cannellini beans, white balsamic vinegar, 2 tbsp olive oil, water, Dijon mustard, and 81 tsp salt) in a small blender. Blend on high until the mixture is completely smooth and creamy, about 30-45 seconds.
Build the Salad Base: Using a mandoline or sharp knife, slice the cucumber into very thin rounds. Spread the arugula evenly across a large platter to create a bed of greens. Arrange the thinly sliced cucumber rounds over the arugula.
Assemble the Salad: Slice the heirloom tomatoes into 41-inch thick rounds. Arrange the tomato slices and the marinated tofu medallions on the bed of greens, alternating them in an overlapping pattern.
Dress and Garnish: Drizzle the creamy white bean vinaigrette generously over the arranged tomato and tofu. Tuck the fresh basil leaves in between the slices. Finish with a few turns of freshly ground black pepper and serve immediately.