In a medium pot combine 1 can refried beans, 8 oz Velveeta, and 1/2 can RO*TEL. Heat over low–medium, stirring frequently, until the cheese is fully melted and the mixture is smooth. Remove from heat.
Warm the 12 tortillas so they are pliable (for example, microwave a stack wrapped in a damp cloth for 20–30 seconds or warm each briefly in a dry skillet).
In a small bowl whisk the 2 egg whites until slightly frothy.
Divide the bean-and-cheese filling evenly among the 12 warmed tortillas. Place the filling on the lower third of each tortilla.
Sprinkle Monterrey Jack cheese over the filling on each tortilla.
Brush or dab a thin edge of each tortilla with egg white to help seal. Fold each tortilla by folding the bottom edge up over the filling, folding both sides inward, then rolling tightly away from you so the seam ends up on the bottom. Use a little more egg white on the final seam if needed to seal.
Preheat the oven to 350°F.
Heat the 2 tbsp vegetable oil in a large skillet over medium heat until shimmering.
Working in batches so you do not crowd the pan, place chimichangas seam-side down in the hot oil and cook until golden, about 1 minute per side, turning as needed to brown all sides.
Transfer the fried chimichangas seam-side down to a baking sheet or ovenproof pan.
Bake in the preheated 350°F oven for 12–15 minutes, until heated through and crisp.
Remove from the oven and let rest briefly before serving.