Gather all ingredients.

Pour olive oil into a saucepan; add onion, poblano, garlic, salt, pepper, chipotle, cumin, oregano, cayenne, and cooked chicken, pressing down as you go. Add currants.

Place saucepan over high heat; cook until onions begin to soften slightly, about 3 minutes. Stir in diced tomatoes with green chiles. Pour in ⅓ cup water and mix to combine.

Bring mixture to a simmer; reduce heat to medium and cook, stirring occasionally, until liquids are cooked down, 30 to 45 minutes. Remove from heat and let cool to room temperature, about 30 minutes. Chill thoroughly before use, 1 to 2 hours.

While filling simmers, combine flour, salt, and sugar for dough in a bowl with your hand. Grate in cold butter, pausing as needed to combine.

Drizzle in beaten egg and water. Mix with a fork until it starts to form a shaggy dough.

Use your hands to squeeze mixture into a lump of dough; transfer to a work surface and knead into a ball of dough, 10 to 15 seconds. Wrap dough in plastic wrap and refrigerate for at least 1 hour, up to overnight.

Preheat the oven to 400 degrees F (200 degrees C). Line a baking sheet with parchment paper. Unwrap chilled dough and transfer to a lightly floured work surface. Use a bench scraper to divide into 8 equal pieces. Roll each piece into an 8-inch circle.

Drain any excess juice from filling. Top each dough circle with a rounded ½-cup measure of the chicken mixture. Sprinkle some pepper Jack cheese over top.

Brush edges of each dough circle with a small amount of water to dampen; fold over and seal with fingertips, making sure to not trap in any pockets of air. Crimp using your fingertips or a fork.

Combine beaten egg and 2 teaspoons water for egg wash. Transfer empanadas to the prepared baking sheet and brush over egg wash. Sprinkle with cayenne.

Bake in the center of the preheated oven until browned and crispy on the edges, about 35 minutes. Transfer to a wire rack to cool for about 5 minutes before serving.

Serve warm and enjoy!