Preheat the broiler. Broil the frying pepper (or bell pepper) on a broiler pan, turning occasionally, until blackened all over (alternatively, char over a gas flame on the stovetop). Transfer to a bowl, cover and let cool for a couple of minutes, then peel away charred skin. Slice the peppers.
In a bowl, combine the chopped chile and vinegar. Season with salt.
In a large skillet, heat a drizzle of EVOO over medium-high heat.Form the sausage meat into 4 thin patties and cook through in the skillet, turning once, 6 to 7 minutes. Transfer to a plate. Wipe out the skillet and reserve.
In a large bowl, dress the arugula with the lemon juice and 2 tbsp. EVOO; season with salt and pepper. Top with the cheese.
Cut a 1 1/2-inch hole in the center of the bread slices. In the reserved skillet, heat the remaining 2 tbsp. EVOO and the butter over medium heat. Add the garlic and cook, stirring, for 1 minute; discard the garlic. Add the bread; season with salt. Fry until golden, then flip.
Crack an egg into each bread hole. Top with a few drops of the chile vinegar and a few slices of roasted pepper strapped across the raw eggs. Loosely cover the pan with foil to set the eggs, about 1 minute. Flip the toast again and cook for 1 minute, then flip back, pepper side up.
To serve, slide the egg toasts onto plates. Serve with a sausage patty and the arugula salad.