Place the diced persimmon in a medium-sized bowl. Add lime juice and basil. Stir well. Season with salt and pepper to taste. Set the salsa aside for up to 2 hours.
Cut the Brie cheese into 1/2-inch slices and then cut each slice into 2-3 squares (depending on the length of the slice). You should have about 32 squares of Brie. In a small bowl, beat the egg and, in a separate bowl, place the panko breadcrumbs. Dip each piece of Brie cheese into the egg, letting the excess drip off and then press into the breadcrumbs, ensuring that both sides of cheese is covered by breadcrumbs. Place on a plate and repeat the process with the remaining pieces of cheese. Cover and chill until ready to cook the cheese. This can be done up to 1 hour in advance.
Just before cooking the Brie, place the crackers on a serving platter. Heat 1 teaspoon olive oil in a medium nonstick skillet over medium-high heat. Add about 1/3 of the breaded Brie pieces to the pan. Cook for about 30 seconds to 1 minute, or until the bottoms of the cheese pieces are light golden brown. Using two forks, flip the pieces over and let the other side brown slightly, an additional 30 seconds to 1 minute. Immediately remove from the pan and place the Brie pieces on the crackers. Repeat with the remaining olive oil and breaded Brie pieces.
Top each piece of Brie with 2 to 3 teaspoons of the persimmon salsa. Serve.