Uncover the pan, raise the heat to medium and stir in salt — I start with 1 teaspoon kosher salt. Cook onions, stirring occasionally, for another 15 minutes. They will not be fully caramelized at this point; it is not what we are going for. Increase heat to medium-high and cook onions until browned at the edges and utterly delicious. Add madeira and vinegar and scrape up any onion bits stuck to the pan. Cook, stirring, until both liquids disappear and the onions are dark. Transfer onions to a large bowl. If you’d like to save a little for garnish, you can set aside a couple tablespoons of them now.
Cook the livers: Add 3 more tablespoons schmaltz to the empty pan and heat over medium-high. Add the livers in one layer and season very well with salt and pepper. Cook for 3 minutes, until lightly browned underneath and flip the livers, seasoning again with salt and pepper, and browning them on the second side, about 2 minutes.
Add the livers to the bowl with the onions, pour the last 1 to 2 tablespoons of schmaltz over, and let everything cool completely. If you’re getting an advance on the liver, I vote for fully chilling them in the fridge overnight. I find that pate blends much more smoothly and light when everything is cold.
To finish and serve: In a food processor, blend the liver and onions until absolutely smooth and as whipped as you can get it. Taste for seasoning; I almost always need more salt and pepper. Transfer to a serving bowl. If you’ve reserved cook onions, you can scatter them on top. If not, a drizzle of oil and some herbs works too. Serve with crackers and garnishes of your choice.
Do ahead: Leftover prepared liver keeps in the fridge for 3 to 4 days. I often make it up to a week before I need it and freeze it. Defrost in the fridge for 24. If you’ve got time, I sometimes re-blend it for a lighter texture once defrosted.
A few onion tips: Use the yellow onions with brown skins if you can get them. If you’re in doubt whether it’s big enough, add another. I used a sweeter Spanish-ish variety for one batch and I’m always bummed the onion flavor isn’t as present and that they’re sometimes so wet, it feels like they turn to mush instead of caramelizing. I always start with an onion or two more than I need, because due to the vagaries of buying onions from grocery stores in the middle of winter, I never know when I’ll get one kind of banged up inside, except reliably any time I don’t buy extras.