In a bowl, whisk together the flour, baking powder, salt, and pepper to taste. In a separate bowl, whisk together the eggs and oil until just combined.
Make a hole in the center of the flour mixture and pour in the egg mixture. Using a rubber spatula, stir everything together until thoroughly incorporated. The batter will be very thick and sticky. Cover with plastic wrap and refrigerate for 3 hours. Fill a large pot with a lid 3/4 full of water and bring to a simmer. In another large pot, bring the stock to a simmer, cover, and turn the heat to low.
While the water is heating, remove the matzo ball batter from the refrigerator. Take about 2 teaspoons’ worth of batter (roughly 20 to 22 grams) and, with wet hands, roll the dough between your palms to make balls. [Editor's Note: The matzo balls may look teensy but they'll increase dramatically in size when added to the water.] Repeat with the remaining batter.
Bring the simmering water to a boil. Gently drop half the matzo balls into the water; when the balls rise to the surface, turn the heat down to a simmer and cover the pot. Cook for 20 to 22 minutes, until the matzo balls are cooked through and the centers are light. If the center is hard and dark, cook until the center is cooked and light, up to 3 or 5 minutes more.
You'll want to handle the matzo balls carefully as they're a touch more delicate than your usual matzo balls. Transfer the cooked matzo balls to the pot of warm broth and repeat with the remaining matzo balls and the simmering water.
Bring the broth with the matzo balls to a simmer. Ladle the soup into bowls, allowing 1 to 2 matzo balls per serving, and garnish with dill.