Preheat your oven to 400°F (200°C). Lightly grease a 9-inch pie dish or similar baking dish and set aside.
In a large skillet, heat the olive oil and butter over medium heat until the butter is melted and foamy.
Add the diced onion and cook for 3 to 4 minutes, stirring often, until softened and translucent.
Stir in the minced garlic and chopped jalapeños, cooking for 1 to 2 minutes until fragrant but not browned.
Add the diced carrots to the skillet and cook for another 3 to 4 minutes, just until they begin to soften.
Sprinkle the flour evenly over the vegetables, stirring constantly for 1 to 2 minutes to form a thick paste and cook off the raw flour taste.
Gradually pour in the beef broth while stirring, making sure to break up any lumps, and cook for 2 to 3 minutes until slightly thickened.
Pour in the heavy cream, stirring to combine, and bring the mixture to a gentle simmer for 2 to 3 minutes until it thickens into a creamy sauce.
Season the sauce with kosher salt, black pepper, smoked paprika, and dried thyme, stirring well to distribute the spices evenly.
Fold in the shredded brisket and frozen peas, mixing until everything is evenly coated in the sauce and heated through, about 2 minutes.
Remove the skillet from the heat and stir in the shredded sharp cheddar cheese and Monterey Jack cheese until melted and smooth.
Taste the filling and adjust seasoning with additional salt or pepper if needed, then transfer the mixture into the prepared pie dish, smoothing the top.
On a lightly floured surface, gently roll out the puff pastry sheet to fit your baking dish, then lay it over the filling, tucking or crimping the edges as desired.
Use a sharp knife to cut 3 to 4 small slits in the center of the pastry to allow steam to escape while baking.
In a small bowl, whisk together the egg and water to make an egg wash, then brush it evenly over the puff pastry surface.
Place the dish on a baking sheet to catch any drips and bake in the preheated oven for 25 to 30 minutes, or until the pastry is puffed and deep golden brown.
Remove the pot pie from the oven and let it rest for 10 to 15 minutes so the filling can set slightly before serving.
Spoon the brisket jalapeño and cheese pot pie into bowls or plates and serve warm, enjoying the creamy, cheesy filling and flaky crust.