Boil chicken: Place the chicken, measured water, 3/4 tsp salt, 1/2 tsp black pepper, and garlic clove in a medium saucepan. Bring to a boil over medium-high heat, and allow to boil, uncovered, until the chicken is cooked through and the water completely evaporates (~25 min). If the water evaporates too quickly, add a little water until it cooks through. If the chicken cooks and starts shedding while water remains, increase the heat to evaporate remaining water. Turn off the heat.
Shred: Transfer the chicken with any leftover juices to an electric mixer bowl with a paddle attachment. Turn the mixer ON to low-medium speed (4). Mix until the chicken is shredded and no large chunks remain (~1-2 min). Alternatively, finely shred on a cutting board using 2 forks.
Make filling: Wipe the saucepan clean and place it back over medium heat. Add the butter and oil. Once the butter melts, add the cumin seeds (if using), onion, and green chili pepper. Sauté until translucent (~6 min), taking care not to brown the onion. Season with the remaining 3/4 tsp salt and 1/2 tsp black pepper.
Finish filling: Add flour and cook, stirring, for 1 minute to remove the floury taste. Pour in the milk and heavy whipping cream and bring to a gentle simmer (~1-2 min). Once it starts to thicken, turn off the heat and add the shredded chicken. Stir to combine. Mix in cilantro, if using. Taste and add more pepper, if you'd like. (Be generous with the seasoning because it will mellow out within the pastries.) Cover and allow to completely cool. You can place in the fridge to speed up chilling.
Prep oven: Preheat oven to 375°F/190°C and place 2 oven racks toward the center of the oven. Line 2 baking sheets with parchment or aluminum foil.
Make egg wash: In a small bowl, whisk egg with water/milk.
Prep pastry sheets: Unfold and lay 1 box of pastry sheets flat on a parchment-lined surface or directly on your baking sheet (keep the other box refrigerated). Use a pastry brush to brush the top of the sheets with the egg wash. Use a sharp knife or pizza cutter to gently cut (try not to slide the knife down) the pastry sheets into approximately 3.25" X 3.25" squares. (See Note 2)
Add filling: Use a cookie scoop to add about 1 tbsp of the chicken to each square. Then close the pastry by folding one edge of the square over to the opposite edge, distributing the chicken toward the edges as needed. Seal the edges by indenting with a fork. Brush the top of each pastry with a light coat of egg wash. Repeat with remaining pastries.
Bake: Place both baking sheets toward the center of your oven (keep space between them). Bake for 30 minutes, or until the pastries on the top sheet are golden brown and flaky. (Note 3) Remove the top tray from the oven and shift the bottom tray to the top rack until it's also golden and flaky (~3-5 min). Serve warm (preferable) or at room temperature. (See note on Serving).