Arrange racks in upper and lower thirds of oven; preheat to 400°F/200°C. Line 2 rimmed baking sheets with parchment paper.

Stir vinegar, honey, red pepper flakes, salt, pepper, and 1 cup water in a medium pot. Add the halved figs and bring to a boil. Reduce heat and simmer until figs are softened and liquid has almost evaporated and is thick and syrupy, 7–10 minutes. Transfer figs to a plate; let cool.

In a separate bowl, combine the feta, cream cheese or goat cheese and oregano with a fork.

Beat egg with 1 Tbsp. water in a small bowl. Roll each sheet of puff pastry to a 12x9" rectangle. Cut pastry along the 12" side into 12 (1-inch-wide) strips. Cut each strip crosswise into pieces about 4 1/2" long. You should have 48 (4 1/2x1") strips pastry.

Scoop a heaping 1/2 tsp. of cheese mixture into cut side of 1 fig half and press so that it adheres. Place fig half in the center of 1 pastry strip. Brush 1 end of pastry with egg wash and roll to enclose. Transfer seam side down to prepared sheet. Repeat with remaining cheese, figs, and pastry. Brush top of each pastry with egg wash and sprinkle with crushed pistachios.

Bake pastries, rotating and switching sheets top to bottom halfway through, until golden brown and puffed, 15–18 minutes.