Add the halved cherries to a small saucepan with ¼ cup granulated sugar and ¼ cup amaretto liqueur. Bring to a boil. Reduce the heat to low and simmer for 15 minutes. Remove from the heat.
Strain the syrup from the cherries into a heat-proof measuring glass. Transfer the cherries to a bowl. Transfer the measuring glass of syrup and the bowl of cherries to the refrigerator for 30 minutes.
Prepare the ricotta mixture:
Gather 1 cup of ricotta cheese, 4 ounces of cream cheese, ¼ cup of granulated sugar, and 1 egg in a large mixing bowl. Add ¼ cup of the reserved amaretto cherry syrup. Beat with an electric mixer until smooth. Add ¼ cup of sifted flour and mix until combined. It’s okay if a few lumps remain.
Prepare the egg wash:
Crack the remaining egg into a small bowl. Whisk with 1 tablespoon water until smooth. Set aside.
Assemble the tart:
Preheat the oven to 375ºF.
Remove the thawed puff pastry from the refrigerator. Line a baking sheet with parchment paper. Place the sheet of puff pastry on the lined baking sheet.
Carefully pinch ¼ inch of the edges of the sheet of puff pastry to create a lip around the edge. Brush the edges of the puff pastry with the egg wash.
Use a fork to prick the inside of the puff pastry lightly.
Spoon the ricotta mixture onto the puff pastry, leaving about ½ inch of space between the ricotta and the inside edge of the pastry.
Arrange the quartered figs and the halved cherries on top of the ricotta. Transfer to the oven for 30 minutes or until the pastry is golden brown around the edges and the ricotta has set.
To serve:
Remove the tart from the oven and let it stand at room temperature for 30 minutes. Slice into pieces and serve with crushed pistachios and fresh mint on top. Finish each slice with a small drizzle of honey if desired.