In a large mixing bowl, whisk the flour, cream of tartar, baking soda, and salt together; set aside.
In the bowl of a stand mixer fitted with a paddle attachment, mix the butter and cream cheese together until smooth and creamy. Add the 1/2 cup granulated sugar and powdered sugar, beating until combined. Add the vegetable oil, egg, lemon juice, lemon zest, vanilla, and almond extract and mix until fully incorporated.
Gradually mix flour mixture into wet ingredients until dough comes together. Wrap dough in plastic wrap and chill in the refrigerator for at least 1 hour.
Preheat the oven to 350ºF. Line a baking sheet with parchment paper or spray with nonstick cooking spray.
Place remaining 1/4 cup granulated sugar into a small bowl. Roll dough into balls that are a little larger than golf balls. Coat ball in sugar and place on lined baking sheet. Repeat process with more dough until you have 6 or 7 balls spaced at least 2 inches apart. Using the bottom of a glass, flatten each dough ball to about 1/2" thickness. It's okay if the cookies form a rim on the edges.
Bake cookies for 8-10 minutes or until tops look dry. They should still be pale in color--no browning. Try not to overbake or you'll end up with hard, crispy cookies. Remove from the oven and allow to cool on the baking sheet for 3-4 minutes before transferring to a wire rack to cool completely.
Make the Frostings:
In the bowl of a stand mixer fitted with a paddle attachment, beat the butter, salt, vanilla, and almond extract together on high until creamy, about 2 minutes. Gradually add the powdered sugar, a cup at a time, beating until frosting is smooth and fluffy.
Transfer half of the frosting to another the bowl; set aside. To the frosting in the mixer, add the jam, 1.5 teaspoons of the lemon juice, and food coloring. Mix until smooth and creamy. Stir the remaining 1.5 Tablespoons lemon juice into the other half of the frosting.
To Decorate:
Drop a tablespoon of lemon frosting onto one half of the cookie, and a tablespoon of strawberry frosting onto the other half of the cookie. Using an offset spatula or butter knife, spread/blend the two frostings into an even layer.
Notes
*If you can't find seedless jam, just press the jam through a fine mesh sieve.