Preheat the oven to 400 degrees F and line two baking sheets with parchment paper. Place the softened goat cheese, 1 egg, honey and salt in a small bowl. Mash together with a fork until smooth.
Lay the puff pastry sheets out on a floured work surface. Roll them slightly to remove the creases. Use a large knife or pizza cutter to cut each piece into 4 equal squares. Then use a fork to prick the center of each square, leaving a 1/2-inch border around the edges.
Spoon the goat cheese mixture evenly on the 8 squares. Use a spatula to smooth out the goat cheese, covering the fork pricks. Then gently fold the borders over the edges of the goat cheese, pinching the corners to make smaller squares. Use the fork to gently crimp the flat edges on the sides of all the tarts. Then move them to the prepared baking sheets.
Whisk the remaining egg with 2 tablespoons of water. Brush the egg wash over the crimped crusts. Bake the tarts for 15-20 minutes until the edges are golden brown.
Remove the tarts from the oven and cool for at least 10 minutes. Then top with small slices of honeydew and fresh blackberries. Drizzle with honey and serve warm or at room temperature.