Preheat the oven to 375°F/190°C and line a baking sheet with parchment paper or a silicone baking mat. Make sure the oven rack is in the middle of the oven.
Make the dough. In a large bowl with a hand mixer, or in a stand mixer with the paddle attachment, begin to mix the vegan butter and sugar together. Add in the applesauce and vanilla extract, mixing again. Sift the flour and cornstarch together before adding this to the wet ingredients. Mix well on low speed until the ingredients come together, then mix on medium speed until the dough forms.
Shape and fill the cookies. Using a cookie scoop, scoop the dough and roll the dough ball in your hands. Roll the cookie dough ball into a small bowl of sugar until the dough ball is fully covered. Transfer to the prepared cookie sheet. Using your finger or the back of a small teaspoon, create a divot in the cookie. Fill with a little less than 1 teaspoon of the jam of your choice. Repeat until all the cookies are made.
Chill. Let the cookies chill for 10 minutes while the oven is preheating.
Bake. Bake the cookies for 17 minutes. If you are baking them on the same sheet you have chilled them on, you will need to accommodate for the cold tray, making the baking time closer to 22 to 24 minutes. Cookies are done when they are set and slightly golden brown in color. Be careful not to tip them over while hot from the oven to avoid burning yourself with hot jam!
Cool and serve. Let the cookies cool for at least 15 minutes before serving. Enjoy!