Combine 2 1/4 cups plus 1 tablespoon flour, ginger, baking powder, cinnamon, baking soda, allspice, nutmeg, and ground cloves in a bowl. Whisk to combine, then set aside.
Combine cream cheese and butter in the bowl of a stand mixer. Blend on medium-high speed until smooth and creamy. Add white and brown sugar, molasses, vanilla extract, almond extract, and salt. Blend on medium-high speed until smooth. Add in egg and egg yolk and blend again on medium speed, 1 to 2 minutes. Scrape down the sides as necessary.
Add the dry mixture slowly just until dough is blended, scraping down sides as necessary. Dough will be very soft. Cover work bowl with plastic wrap and refrigerate for at least 2 hours, up to overnight, to allow dough to chill and firm.
Preheat the oven to 325 degrees F (165 degrees C). Line baking sheets with Silpat® mat or parchment paper.
Roll dough into approximately walnut-sized balls. Pour confectioners' sugar into a bowl and roll dough balls in the sugar until very well coated. Set on the prepared baking sheet, leaving 1 1/2 to 2 inches of space in between, about 8 to 12 cookies per sheet, depending on how large you decide to make the cookies.
Bake in the preheated oven until the edges are just starting to firm and become golden browned and the center looks mostly set, 15 to 17 minutes. Cool on baking sheet for 2 minutes before carefully transferring to a wire rack to cool. This should yield a chewy cookie with slightly crispy edges. Repeat with remaining cookie dough.