Arrange the tomato slices in a shallow dish in a single layer. In a small bowl, whisk together the balsamic vinegar, water, salt, and 1/8 teaspoon pepper. Pour over the tomato slices and set aside.
In the bowl used for the vinegar mixture, combine the mayonnaise and pesto. Set aside.
In a large bowl, combine the plant-based ground beef, Parmesan, tomato paste, salt, and remaining 1/2 teaspoon pepper. Divide the mixture into four equal portions. Roll each portion into a ball, then shape into patties.
Heat a grill, skillet, or grill pan over medium-high heat. Add the burger patties and cook for 4 minutes. Flip the burgers over and add mozzarella slices. Cook patties for an additional 4 minutes, until the cheese is softened and the internal temperature of the burgers reaches 160°F. Remove from heat and set aside.
Place the buns face-side down onto the heat source and toast for 1-2 minutes.
Spread the pesto mayo on the bottom halves of the buns. Top each with a burger patty, marinated tomato slices, fresh basil leaves, and the top bun. Serve immediately.