Preheat the oven to 400 degrees F (200 degrees C). Line baking sheets with aluminum foil.

Make 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 to 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 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 to 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 of the bottom of the pan with a wooden spoon. Pour liquid from the skillet over 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 a skillet over medium-high heat; sauté onion 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 remaining 1 tablespoon olive oil. Flip slices over so the oiled sides face downward. Spread cheese sauce generously onto each slice. Spoon beef mixture on top. Sprinkle with provolone cheese.

Bake in the preheated oven until browned and cheese is melted, 12 to 15 minutes.