Heat a mid sized skillet over medium heat. Brush either side of sourdough with 1 Tbsp oil. Place bread face down and toast until slightly golden, 3-4 minutes.
Keeping skillet at medium heat add 1 tsp of hatch chili oil and cook egg, season with Chili Verde Sea Salt. Once whites are almost set add cheese and cover pan for 1-2 minutes to let cheese melt.
While the egg is cooking smear one side of the bread with red pepper jam. Layer with prosciutto and arugula.
Carefully scoop egg and cheese onto the sandwich and close with the second piece of bread. Serve immediately.