Bring a large pot of water to a boil. Add potatoes; cook until mostly tender, about 5 minutes. Add peas; continue cooking until potatoes are soft, 5 to 10 minutes. Drain.
Melt butter in a large skillet over medium heat. Season ground beef with salt and pepper and add to the skillet with onion; cook and stir until beef is browned and onion is translucent, about 5 minutes.
Pour 2 1/2 cups water over beef mixture; bring to a boil. Reduce heat to medium and stir in potatoes and peas. Simmer, skimming scum off the surface, until flavors combine, about 15 minutes. Remove from heat and let cool, about 5 minutes. Add curry roux cubes; stir until dissolved.
Let curry mixture cool, about 20 minutes. Cover with plastic wrap and refrigerate until thickened, 8 hours to overnight.
Heat oil in a deep-fryer or large saucepan to 350 degrees F (175 degrees C).
Cut dough into 1/2-inch slices and roll out each slice to a 3-inch circle. Place a teaspoon of chilled curry into the center of each circle. Pull up sides of the circle over curry and pinch edges together to form a round ball.
Beat egg in a small bowl. Place panko bread crumbs in a shallow dish. Dip each ball of dough into the beaten egg. Press into bread crumbs to coat, shaking off excess.
Fry in the hot oil until evenly browned on all sides, 3 to 5 minutes.
In step 2, cover beef with the amount of water specified on the package of curry roux you are using.
We have determined the nutritional value of oil for frying based on a retention value of 10% after cooking. Amount will vary depending on cooking time and temperature, ingredient density, and specific type of oil used.
Parchment can be used for easier cleanup/removal from the pan.