In a medium mixing bowl, beat the eggs with a fork and then add the peppers, tomatoes, feta cheese, dill, salt, and pepper. Fold ingredients in until thoroughly combined. Set aside.
Cut the brioche buns in half. For each half, scoop out the center until the bun resembles a small cup. Pinch the sides and press firmly on the center to avoid loose crumbs. Place each cup on a non-stick baking pan.
Add the egg mixture to the buns using a ladle. The buns will absorb some, so you may have to come back and add some more to fill them to the top.
Leaving the pan uncovered, bake the veggie egg cups for 10 minutes at 350ºF, just so the eggs set.
Place foil on top and bake for 15 more minutes. Serve immediately.