Heat oven to 375 with a rack in the upper third of the oven. Oil a 9- by 13-in. baking dish with 1 tbsp. oil. Drop 1/2-cup scoops of mashed potatoes into dish, spaced about 2 in. apart.
Heat remaining 1 tbsp. oil in a large frying pan over high heat. Cook sausage, turning often, until golden brown, about 5 minutes. Scatter cooked sausage in pan, laying over and between potato mounds.
Add spinach to frying pan and wilt, stirring occasionally, about 1 minute.
Sprinkle spinach over sausage.
Whisk eggs together in a large bowl until broken up.
Whisk in flour, baking powder, salt, thyme, and onions.
Pour mixture over sausage and potatoes.
Sprinkle cheese on top.
Bake until potatoes are bubbling, 20 to 25 minutes.
