Add flour, yeast, sugar, and salt to the bowl of a stand mixer. Using the dough hook, mix the ingredients on low speed while adding the olive oil. Being adding the warm water with the mixer on low speed and add just enough to have the dough begin to form a dough ball. Continue adding 1 Tablespoon of water at a time until the dough releases from the bowl and is just barely sticking at the very bottom under the dough hook.
Remove the dough from the bowl of the mixer. It should feel tacky and stick just a bit to your fingers. Drizzle some olive oil into a glass bowl and coat the dough ball. Cover with plastic wrap and let rise on the counter for about 60 minutes or until it doubles in size. OR, if you have the proof function on the Ninja Foodi, you can use that on 95°F/35°C for 30-45 minutes.
Deflate the dough and shape it into a round disc. Wrap in plastic wrap and refrigerate if you are not using it right away. Or proceed to rolling out the dough.
Egg Filling
Heat 1 Tablespoon of butter in a large skillet (about 10") on medium heat. Blend 6 eggs with salt and pepper in a blender. Add ¼ cream and pulse blend until well combined.
Once the butter has melted, pour egg mixture into heated pan. Cover and turn the heat on low. Cook on low heat for about 10-15 minutes or until the eggs are completely set. Turn off the heat and allow the eggs to cool for about 15 minutes before using in the stromboli.
Sausage & Vegetable Filling
Add about 1 Tablespoon of butter to a large sauté pan and turn the heat on medium high. When the butter has melted add in the strips of onion and peppers. Season with salt and pepper.
Cook for about 2-3 minutes and then add the sausage. Continue to cook, stirring occasionally until the sausage is cooked through and the vegetables are the texture you like. This will take about 10 minutes usually.
Remove the mixture from the pan and put into a large bowl or measuring cup. You will have about 4 cups of the mixture. Let sit so the juices and fat collect in the bottom of the bowl/measuring cup.
Cheese Sauce
Remove about all but 2 cups of the sausage and vegetable mixture and put it in a high speed blender. Add 1 cup of milk and blend until smooth.
In a large skillet, add the juices from the sausage and pepper mix. You should have about 3 Tablespoons OR you can add 1 Tablespoon of butter if you aren't sure if you have enough fat in the liquid.
Add in 1 Tablespoon of flour and heat over medium heat while stirring until a paste forms. Add in the blended milk and vegetables and stir until smooth. Add the shredded cheese and stir until melted. If the sauce is too thick, add some more milk. If it is too thin, you can add more cheese or whisk together 1 Tablespoon of water with ½ Tablespoon of cornstarch and stir the slurry into the sauce. Bring to a simmer to activate the cornstarch and the sauce will thicken.
Assembling the Stromboli
If you haven't already, preheat the oven to 425°F/ 220°C. Add the diced apple to the sausage and pepper mixture and stir to combine so the apples don't discolor.
Lightly flour a large work surface. Shape your dough ball into a rectangle and roll it out with a rolling pin so it is about 14"x20".
Place the dough on a piece of parchment paper. Cut the large egg round in half and place them about 3-4" from the long edge of the dough closest to you. You want the egg to extend almost edge to edge, but it may overlap in the middle. Spread a thin layer of sauce over the eggs, you will use about ½ cup of the sauce. Top with the remaining 2 cups of sausage and pepper mixture. Top with the cheese strips.
Begin rolling the stromboli, starting at the edge closest to you and bringing the dough up and over the filling. Close in the sides of the stromboli by pulling the sides of the dough from each side over to meet the dough on top.
Gently roll the stromboli over until you have about 3-4" left of the dough furthest from you. Grab the long edge of the dough and pull it up and over to completely encase the filling in dough. Press the seams together to form a seal.
Roll the stromboli so the seam side is down on the parchment paper and cut a few slits in the top. Brush with olive oil and sprinkle with salt. Place the parchment paper and stromboli onto a large baking sheet pan.
Baking the Stromboli
Place the stromboli into the preheated oven and bake for about 25 minutes or until it is golden brown.
Allow to cool on a cooling rack for about 5-10 minutes. Cut slices with a serrated knife and serve with extra cheese sauce. Enjoy!