Place the Swiss chard in a large saucepan with enough water to cover it, add some salt, and bring it to a boil over high heat.
Reduce the heat to medium and cook the chard until it becomes tender, which takes about 15 minutes.
Drain the chard and finely chop it.
Heat 1 tablespoon of olive oil in a medium skillet over medium heat.
Add the chopped onion and cook until it becomes wilted, which takes about 5 minutes.
Add the chopped chard, salt, pepper, and oregano, and continue to cook until the chard wilts and absorbs the seasonings.
Meanwhile, heat the butter and the remaining 2 tablespoons of olive oil in a medium saucepan over medium heat until the butter melts.
Add the flour and cook, stirring constantly, until the sauce becomes thickened and smooth, approximately 3 minutes.
Add the chard and continue to cook, stirring occasionally, for an additional 10 minutes.
Spread the mixture onto a dinner plate and allow it to cool, then refrigerate until it's firm enough to handle.
Shape the chard mixture into 1-inch balls and dip them in the beaten eggs, followed by coating them with the bread crumbs.
Place the eggs and bread crumbs in separate shallow bowls.
Heat at least 1 inch of olive oil in a large skillet over medium-high heat, or, for better results, use a deep fryer set at 365°F.
The oil should be hot enough that a cube of bread turns light brown in 60 seconds.
Dip each croquette in the egg, then dredge it in the bread crumbs, and gently place it in the hot oil.
Fry the croquettes, turning them several times, until they become golden.
Use a slotted spoon to transfer the fried Swiss chard croquettes to paper towels to drain any excess oil.