In a 10-inch cast-iron or straight-sided pan, heat the olive oil over medium-high heat. Add the sausage pieces and break apart further with a wooden or firm spoon. Brown the sausage for about 4 minutes. Reduce the heat to medium and add the onion and garlic. Cook to let soften, about 2 minutes.
Add the mushrooms and return the heat to medium-high. Cook about 4 minutes, then add the Swiss chard. Season with salt and pepper.
Continue to cook the mixture until the Swiss chard is fully softened, about 6 minutes longer. Stir in the parsley.
Turn off the heat and fold in the eggs. Season with salt and pepper.
Place the pieces of Taleggio on top of the eggs and transfer to the oven to bake just until set and the Taleggio is melted and bubbling, about 10 minutes.