Make the jam: In a small saucepan over medium heat, cook strawberries, sugar, and water until berries soften and release juices.
Stir cornstarch with a small splash of water, then add to the saucepan. Continue cooking until thickened into a jam. Set aside to cool completely.
Preheat oven to 350°F and line a baking sheet with parchment paper.
In a large bowl, cream butter and sugar until smooth and fluffy.
Mix in 3 tbsp cooled strawberry jam, applesauce, and vanilla until fully combined.
Add flour and baking powder and mix just until no dry streaks remain. Do not overmix.
Scoop dough onto baking sheet.
Spoon about 2 tbsp reserved jam over the tops of each cookie.
Use a butter knife to gently swirl the jam into the top of the dough.
Bake for 10–13 minutes, or until edges are just set and lightly golden.
Let cookies cool completely on the baking sheet before moving.