Pour oil into a 10- to 12-inch frying pan over medium-high heat. Add onion and stir often until limp and slightly golden, about 5 minutes. Stir in garlic and crumble sausage into pan. Stir often until meat is no longer pink, 7 to 10 minutes, using a wooden spoon to break up the meat if necessary.
Add escarole to sausage mixture in pan and reduce heat to low. Stir until escarole is wilted, 1 to 2 minutes. Remove pan from heat.
In a large bowl, whisk the milk and eggs until combined. Stir in Gruyère, bread cubes, salt, pepper, and nutmeg.
Add sausage mixture and stir to mix well. Scrape mixture into a buttered 9- by 13-inch baking dish and spread level (see notes).
Bake in a 375° oven until top is golden and a knife inserted in the middle comes out mostly clean, 35 to 45 minutes. (If top is brown after 30 minutes, cover loosely with foil and continue to bake.) Serve hot.