Preheat oven to broil with a rack 2-3 inches from heat source.
Spray a rimmed baking sheet with nonstick spray or brush with olive oil.
Place tortillas on baking sheet and set aside.
To a large nonstick skillet over high heat, add 2 tablespoons olive oil and sausage.
Cook, breaking up into small pieces with a wooden spoon, until beginning to turn golden, about 3 minutes.
Add hash browns and continue to cook, stirring only occasionally, until potatoes are browned and crispy, about 12 minutes.
Add taco seasoning and stir to coat.
Divide mixture among tortillas.
Wipe out skillet and reduce heat to medium low. Add butter and swirl to coat.
When foaming subsides, add eggs and season with salt and pepper.
Using a rubber spatula, gently scramble whites, leaving yolks intact, until nearly cooked, and then crack yolks. Remove from heat and lightly scramble, so that residual heat begins to cook yolks. Undercooked egg is okay, as they will continue to cook under broiler.
Divide eggs among tortillas, along with half of cheddar cheese.
Top each tortilla with one tostada.
Fold tortilla up and around the tostada shell, securing with toothpicks.
Brush with remaining 1 tablespoon olive oil and top with remaining cheddar.
Transfer to oven and broil until tortillas are deeply golden and interior is warmed through, about 3 minutes.
Serve warm, topped with hot sauce and sour cream on the side.