Toast the English muffin halves until lightly browned and set them aside.
Heat the olive oil in a large skillet over medium heat and add the chopped Swiss chard.
Sauté the greens until they are wilted and tender.
Stir in the Dijon mustard, heavy cream, and a pinch of salt and pepper until the liquid has mostly evaporated and coats the greens.
Remove the greens from the pan and drain any excess liquid if necessary.
Divide the greens evenly among the bottom halves of the toasted muffins.
Place a slice of cheddar cheese over the warm greens on each muffin base so it begins to soften.
Wipe the skillet clean with a paper towel and return it to medium heat with the butter.
Crack the eggs into the pan and season them with salt, pepper, and a dusting of smoked paprika.
Fry the eggs until the whites are opaque and firm and the yolks are cooked to your liking.
Place one fried egg on top of each cheese-covered muffin base and close the sandwiches with the remaining muffin tops.