Set a wide saucepan over medium heat and add enough oil to thinly coat the bottom. Add the shallots and salt.
Cover and cook, stirring occasionally, for about 10 minutes, until the shallots have significantly softened and turned a pale, almost-transparent pink.
Remove the lid and drop the heat to medium-low. Stir in the sugar along with a small splash (about 1 tablespoon) of water. Continue cooking for 30 minutes—uncovered, stirring occasionally, and deglazing with water (about 1 tablespoon at a time) if anything starts to stubbornly stick.
Stir in 4 tablespoons of the vinegar. Continue cooking for another 10 to 15 minutes, stirring occasionally, until the texture is as jammy as you’d like. At this point, stir in the remaining vinegar, then cut the heat.
Taste and add more vinegar or salt, if you’d like.
Let cool, then jar and refrigerate for up to a couple weeks. If the jam becomes too stiff once chilled, you can loosen with a bit of water, vinegar, or both.
