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 add the egg mix. Fold together with a rubber spatula until thoroughly incorporated; the batter will be very thick and sticky. Cover with plastic wrap and refrigerate for 3 hours.
Fill a large lidded pot three-quarters full with water and bring to a simmer. Place the broth in another large pot and bring 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. 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 for another 3 to 5 minutes, until the center is cooked and light. Transfer to the warmed broth, and repeat with the remaining matzo balls.
Bring the vegetable broth and matzo balls to a simmer. Serve one to two matzo balls per serving; garnish with dill.