1. Prepare an ice bath. Bring a large pot of water to a boil. Carefully add 9 eggs; boil 6 minutes. Transfer to ice bath until cool enough to handle, then carefully peel and transfer to a paper towel-lined plate.

2. In a medium skillet, heat 1 teaspoon oil over medium-high. Add sausage and cook, stirring to break up, until browned, about 5 minutes. Transfer to a plate; let cool slightly. Chop into pea-size pieces.

3. Preheat oven to 400 degrees. Line 9 cups of a standard 12-cup muffin tin with baking cups, or lightly brush with oil. In a medium bowl, whisk together flour, baking powder, baking soda, salt, and sugar. In another bowl, whisk together remaining 1/2 cup oil, sour cream, 1/4 cup water, and remaining 2 eggs. Stir wet ingredients into dry until just combined. Fold in sausage, scallions, and 1/2 cup cheese. (Batter will be thick and doughlike.)

4. Fill each prepared muffin cup with 2 tablespoons batter. Nestle a cooked egg in each. For each cup, scoop 2 more tablespoons batter and flatten it slightly to form a disk. Cover egg with disk; gently press to seal. Sprinkle tops with remaining 1/2 cup cheese. Bake until golden and tops spring back when lightly touched, about 22 minutes. Let cool 5 minutes in tin, then transfer to a wire rack and let cool completely.