Fill large bowl with ice water. Bring large pot of water to boil. Using slotted spoon, carefully lower 9 eggs into boiling water. Boil 6 minutes, then use slotted spoon to transfer eggs to ice bath. Cool 5 minutes, then carefully peel and transfer to paper towel-lined plate.

In medium skillet over medium high heat, add 1 teaspoon oil. Add sausage and cook, breaking into smaller pieces with wooden spoon or spatula, 5 to 7 minutes until browned. Transfer to plate. Cool 10 minutes, then chop into pea-sized pieces.

Heat oven to 400 degrees F. Line 9 standard-size muffin cups with paper baking liners. In medium bowl, whisk flour, baking powder, baking soda, salt and sugar to combine. In separate medium bowl, whisk remaining 1/2 cup oil, sour cream, 1/4 cup water and remaining 2 eggs. Stir wet ingredients into dry ingredients just until incorporated, then fold in sausage, scallions and 1/2 cup cheese (batter will be very thick).

Spoon 2 tablespoons batter into each prepared muffin cup. Gently press 1 egg into center of batter in each cup. For each cup, scoop 2 more heaping tablespoons batter and form into rough disc; place disc on top of and around egg to cover. Press seams to seal. Sprinkle tops with remaining 1/2 cup cheese.

Bake 21 to 25 minutes until golden brown and tops spring back when lightly touched. Cool 5 minutes in muffin cups, then transfer to cooling rack to cool completely.