1. Preheat the oven to 375°. Brush a large rimmed baking sheet with 2 teaspoons of the olive oil. Lay one sheet of the phyllo dough on the baking sheet. Brush the phyllo with olive oil and sprinkle lightly with breadcrumbs. Repeat the layering with the remaining phyllo, olive oil and breadcrumbs.
2. Bake the crust for about 20-25 minutes, until golden and crisp; rotate the baking sheet halfway through cooking. Let cool completely.
3. In a food processor, pulse the cream cheese with the mayonnaise until smooth. Add a few teaspoons of water and repulse to thin it out as needed – the consistency should allow for it to be very easy to spread. Spread the cream cheese mixture in the center of the tart, leaving a 1” border all around. Arrange the tomato slices on top and sprinkle with flaky sea salt and pepper. Drizzle lightly with olive oil and garnish with basil leaves. Cut into squares and serve at room temperature.