Heat 1 Tbs. of the olive oil in a medium saucepan over medium heat. Add the Swiss chard and cook until wilted and soft, stirring occasionally, 5 to 7 minutes. Season with salt and pepper and set aside to cool slightly.
Heat a panini or sandwich press according to the manufacturer's instructions. Brush both sides of each slice of bread with 1 Tbs. of the olive oil. Place on the press, pull the top down, and cook until toasted, 1 to 2 minutes. Rub one side of each toasted slice with the garlic clove. Spread 1 Tbs. of cream cheese on each slice. Sprinkle with the capers.
Brush the salmon on both sides with the remaining 2 Tbs. olive oil and sprinkle with salt and pepper. Place the salmon on the press, pull the top down, and cook until seared and cooked to your desired degree of doneness, at least 3 minutes.
While the salmon is cooking, heat the butter in a medium skillet. Carefully crack the eggs into the skillet, taking care not to break the yolks. Sprinkle 1 tsp. water onto each egg and sprinkle with salt and pepper. Cover the skillet and cook on medium-high heat until the whites are firm and the yolks are still soft, 2 to 3 minutes.
Arrange the cooked greens on top of the cream cheese, place a salmon fillet on top of each bed of greens, and slide a fried egg on top of each piece of salmon. Sprinkle with salt and pepper. Serve immediately.