Preheat oven to 400°F. Lightly grease a sheet pan.
On a large counter or table, set out 4 squares of aluminum foil that are large enough to hold the tortillas. Top each piece of foil with one tortilla.
Place a large nonstick frying pan over medium heat. When the pan is hot, add 1 Tbsp cooking oil and then bell peppers, and onions with a pinch of salt. Saute the peppers and onions until very tender, 5 to 7 minutes (don’t rush this step because the peppers and onions will develop flavor as they cook).
Add chopped ham and continue cooking until ham starts to brown lightly on the bottom of the pan, 3 to 4 minutes more.
Divide the peppers, onions, and ham between the tortillas.
Return the skillet to the stove and reduce to medium-low heat and add 1 Tbsp cooking oil.
In a small bowl, whisk eggs until evenly combined. Pour the eggs into the bowl and cook, stirring constantly, to scramble the eggs slowly, 5 to 6 minutes. Season eggs with some salt and pepper.
Top each tortilla with 2 slices of cheese and then eggs.
Fold over the sides of the tortillas and then roll them up from the bottom to tightly enclose filling. Wrap tortillas tightly with foil and transfer them to the sheet pan (see below for freezer instructions).
Bake burritos for 10 minutes to let flavors come together and cheese melt.
Optional step: To give the burritos a bit of added texture, peel back the foil on each of the burritos and bake them with the tops exposed to the heat for another 5 minutes. This will give the tops a slightly golden, crisp crust that is super satisfying when you bit into or slice into the burritos.
Serve immediately with guacamole, sour cream, or salsa.