Preheat the oven to 425°F. For the Yorkshire pudding, combine the eggs, flour, milk, and salt in a large mixing bowl and stir everything together until a smooth batter forms. Once combined, let the batter rest for 15 minutes. Slice the red onions into large rings and remove the seeds from the bell peppers and cut them into 4 equally sized rings. Place 1 slice of ham, 1 slice of cheese, and 1 slice of red onion inside each of the bell pepper rings.
Coat a 13-inch oven-safe frying pan with vegetable oil and place it in the oven for 15 minutes. Once the frying pan is hot, carefully remove it from the oven, pour in the Yorkshire pudding batter, and arrange the stuffed bell pepper rings on top. Transfer the frying pan back to the oven and bake the Yorkshire pudding at 425°F for at least 30 minutes, until the Yorkshire pudding has risen.
For the Scotch eggs, soft boil the eggs and season the ground beef with salt & pepper. Peel the soft-boiled eggs, roll them in flour, and cover them in an even layer of ground beef.
Whisk the 2 eggs for the egg wash, fry the bacon, chop it, and heat a pot filled with cooking oil on the stove. Set up the breading station with 1 bowl filled with flour, 1 bowl filled with the egg wash, and 1 bowl filled with bread crumbs and chopped bacon. Cover the coated eggs in flour, dredge them in egg wash, and coat them in the bread crumb mixture, then fry the Scotch eggs in hot cooking oil until golden brown.
Place the Scotch eggs in the stuffed bell pepper rings on top of the Yorkshire pudding in the pan. Pour some béchamel sauce over the Scotch eggs and place some arugula inside the Yorkshire pudding basket.