Add the finely diced apples, brown sugar, cinnamon, and kosher salt to a medium saucepan. Cook over medium-low heat, stirring to combine. Cover and cook for 6–8 minutes, stirring once or twice, until the apples have softened and released some of their juices.
Remove the lid and stir well. Using an immersion blender, carefully puree approximately half of the apple mixture, leaving plenty of small, tender apple pieces intact. The mixture should have a thick, spoonable texture with a combination of smooth jam and soft apple chunks.
Stir in the fresh lemon juice and continue cooking uncovered over medium-low heat for another 2–4 minutes, stirring frequently, until the mixture thickens to a jam-like consistency. The jam should look thick and spreadable but still slightly looser than your desired final consistency, as it will continue thickening as it cools.
Remove the saucepan from the heat and stir in the vanilla bean paste or vanilla extract until evenly combined.
Allow the jam to cool before serving. As it cools, it will continue to thicken. Serve over warm biscuits, toast, pancakes, waffles, French toast, or your favorite baked goods. Transfer leftovers to a clean, airtight container and refrigerate promptly.