In a large saucepan combine apples, 1/2 cup of the water, and the lemon juice. Bring mixture to boiling over medium-high heat, stirring constantly; reduce heat. Simmer, covered, for 25 to 30 minutes or until apples are very tender, stirring frequently. Press apples through a food mill or sieve until you have 5 cups pulp; discard skins.
Meanwhile, for caramel, pour granulated sugar into a 6- to 8-quart heavy pot, spreading evenly. Heat over medium-high heat until sugar begins to melt, shaking pot occasionally; do not stir. When the sugar starts to melt, reduce heat to medium-low and cook for 5 to 10 minutes or until all of the sugar is melted and golden, stirring as necessary with a wooden spoon. Remove from heat. Carefully add the remaining 3/4 cup water (caramel will spatter and become hard). Return to heat; cook and stir over medium heat until caramel is dissolved.
Carefully add the 5 cups apple pulp and brown sugar to caramel (again, caramel will spatter and become hard). Cook over medium heat until brown sugar and caramel are dissolved, stirring constantly. Increase heat to medium-high. Boil gently, uncovered, about 10 minutes or until mixture is thickened, stirring frequently. Remove from heat. Stir in butter and vanilla.
Ladle hot jam into hot sterilized half-pint canning jars, leaving a 1/4-inch headspace. Wipe jar rims; adjust lids and screw bands.
Process filled jars in a boiling-water canner for 10 minutes (start timing when water returns to boiling). Remove jars from canner; cool on wire racks. Makes 6 half-pints.