Step 1

Preheat oven to 375°F. Brush a 2-quart baking dish with butter. Melt butter in a large skillet over medium-high heat. Add onion and 1/2 teaspoon salt; cook, stirring occasionally, until golden in places, 5 to 7 minutes. Add spinach and thyme; cook until spinach wilts and darkens slightly, about 5 minutes. Add potatoes and milk; season with pepper.

Step 2

Simmer until milk thickens and reduces slightly and potatoes are easily pierced with the tip of a knife but still retain their shape, about 5 minutes. Remove from heat; let cool completely. Meanwhile, in a large bowl, whisk eggs with 1 1/2 teaspoons salt and nutmeg. Stir cooled potato mixture and half of cheese into eggs.

Step 3

Transfer to prepared baking dish and cover with parchment-lined foil. (If assembling the day before, wrap the dish with plastic wrap and refrigerate until ready to bake). Bake casserole, covered, 30 minutes. Uncover, sprinkle evenly with remaining half of cheese, and continue baking until set and golden brown in places, 15 to 25 minutes more (if the top is browning too quickly, tent with foil). Let cool slightly before serving.