Position oven racks in the middle and lower third of the oven and preheat to 425 degrees F. Generously spray a 24-cup mini muffin tin with nonstick cooking spray.
Put the hash browns in a medium bowl, add the eggs and toss with a fork to coat. Divide the mixture among the cups of the prepared muffin tin (about 2 tablespoons per cup). Use the back of a tablespoon measure to gently press the hash browns around the bottom and sides to form a thin layer with some shaggy bits sticking up and out of the cups. Sprinkle with salt and pepper.
Bake the hash brown cups on the top rack for 20 minutes, then rotate the muffin tin from front to back. Spread the popcorn shrimp in a single layer on a baking sheet and put on the bottom rack. Continue to bake until the hash brown cups are golden brown on the ends and crispy and the shrimp are warmed through and crispy, about 10 minutes for the shrimp and 10 to 15 minutes for the hash browns.
Meanwhile, melt the butter in a small pot over medium heat. Add the flour and cook, whisking, until it begins to foam, about 2 minutes. Whisk in the warm milk and bring to a simmer; simmer until thickened, 1 to 2 minutes. Remove from the heat and stir in the Cheddar until completely melted into the sauce. Stir in the broccoli and 1/2 teaspoon salt and place over low heat to warm through.
When ready to serve, arrange the hash brown cups on a serving platter. Spoon about 1 heaping tablespoon of the broccoli cheese sauce into the center of each cup. Top with 1 or 2 pieces of popcorn shrimp.
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