Put the chard in a bowl. Heat a large skillet over medium-high heat.
Add 1 tablespoon olive oil and the bread cubes and season with salt and pepper. Cook, tossing, until toasted, 4 to 5 minutes. Add to the chard.
Add the anchovies to the skillet, mashing them with a whisk. Add the garlic and 2 tablespoons olive oil and cook until the garlic is golden, about 45 seconds. Add the tomatoes and warm slightly, then whisk in the lemon juice and season with salt and pepper. Pour the warm dressing over the chard, toss and set aside to wilt.
Wipe out the skillet and place over high heat. Sprinkle the steak on both sides with the mint, and salt and pepper to taste. Add the remaining 1 tablespoon olive oil to the skillet. Sear the steak until browned on the bottom, about 5 minutes. Turn and cook until browned on the other side, 3 to 4 more minutes for medium-rare. Transfer to a cutting board and let rest 5 minutes.
Add the feta to the chard salad and toss. Thinly slice the steak and serve with the chard salad.