Preheat the oven to 400°. Line two baking sheets with aluminum foil.
Make the sauce: Melt butter in a skillet over medium-high heat. Whisk flour into hot butter and cook, whisking constantly, until mixture is pale and flour taste cooks off, about 1 minute. Whisk milk into flour mixture until mixture is hot and thickened, 3-6 minutes. Add provolone cheese, nutmeg, cayenne pepper, and salt; stir until cheese is completely melted. Set aside.
Make mini cheesesteaks: Season steak with salt and black pepper. Heat 1 tablespoon of the olive oil in a skillet over medium-high heat. Cook steak in hot oil, turning occasionally, until meat is slightly firm and pink on the inside, 5-7 minutes. Transfer meat to a plate.
Return the skillet to heat and pour in water; bring to a boil while scraping the browned bits of food off the bottom of the pan with a wooden spoon. Pour liquid from the skillet over the steak. Cool steak to room temperature; dice meat and place meat and accumulated juices from the plate into a large bowl.
Heat 1 tablespoon oil in the skillet over medium-high heat; saute onions and peppers in hot oil until softened and slightly translucent, about 5 minutes. Stir onion and peppers into beef in the bowl; season with salt and pepper.
Spread bread slices out on the prepared baking sheets and drizzle with the remaining 1 tablespoon olive oil. Flip slices over so that the oiled sides face downward. Spread cheese sauce generously onto each slice. Spoon beef mixture on top. Sprinkle with provolone cheese.
Bake in preheated oven until browned and the cheese is melted. 12-15 minutes.