Preheat oven to 400°F Butter a 6x9 or 8x8 inch baking dish.
Heat one of the teaspoons of olive oil in a large skillet over medium-high heat until shimmering. Add spinach and cook, stirring often, until wilted, about 3 minutes. Transfer spinach to strainer and drain. When cool, squeeze out liquid and roughly chop the spinach.
Wipe skillet clean. Add sausage to skillet and cook over medium-high heat until browned, 5 to 6 minutes. Use slotted spoon to transfer sausage to bowl. Wipe skillet clean. Heat remaining oil in skillet and add onions, cooking until golden, about 5 minutes. Add garlic and cook until fragrant, about 1 minute. Scrape mixture into bowl with sausage.
Meanwhile, place potatoes in a large pot and add enough water to cover by 1 inch. Bring potatoes to boil over high heat, then lower heat to a simmer. Cook until potatoes are tender, about 15 minutes. Drain potatoes, wipe pot dry, put potatoes back into the pot, and mash with butter, cream, broth, vinegar, nutmeg, salt and pepper. Stir in spinach and 1/2 cup of the cheese.
Transfer potato-spinach mixture to prepared baking dish. Top with sausage-onion mixture and sprinkle with remaining 1/2 cup of cheese. Bake until potatoes are very hot and cheese is golden and bubbly, 20 to 25 minutes. Cool for 10 minutes before serving.