Toss together the apples, lemon juice, apple pie spice, vanilla and 1/3 cup of the sugar in a medium bowl. Melt 2 tablespoons of the butter in a medium skillet over medium heat. Add the apple mixture and stir to coat with the butter. Cover and cook, stirring occasionally and adding about 1/4 cup water if the mixture becomes too dry or begins to brown, until the apples are tender and shiny, about 20 minutes. When the apples are tender, there should be some liquid left in the skillet; if not, add 1/4 cup water and stir. Mix the cornstarch with 1 teaspoon water in a small bowl and pour it into the apple mixture. Stir well and bring the mixture to a boil to activate the starch. Cook until thick and bubbly, 30 seconds to 1 minute more. Transfer to a bowl and refrigerate for 15 minutes.
Whisk together the flour and salt in a bowl. In another bowl, beat the remaining 8 tablespoons butter and 2/3 cup sugar with an electric mixer on medium-high speed until fluffy, about 5 minutes. Beat in the egg until just combined. Turn the mixer to low and gradually beat in the dry ingredients in 2 additions, beating until just incorporated.
Line 2 baking sheets with parchment. Scoop 1-tablespoon portions of dough and roll between your hands to form 1-inch balls. Place about 2 inches apart on the prepared baking sheets. Gently press a thumbprint into the center of each ball about 1/2 inch deep, being careful not to crack the edges. Fill each indentation with 1 teaspoon of the filling, then chill the cookies 15 minutes in the freezer.
Position oven racks in the top and bottom thirds of the oven and preheat to 350 degrees F.
Bake, rotating the baking sheets from top to bottom and front to back about halfway through, until the cookies are set, about 30 minutes. Cool the cookies completely on the baking sheets. Store the cookies in an airtight container in the refrigerator for up to 5 days.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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