Melt the butter and let it cool down. Add it to a large bowl and whisk in both sugars. Add in the egg and yolk and whisk until combined.
Place the grated carrots in a paper towel and squeeze out all the excess moisture and discard (or drink) the juice. Add the carrots to the mixture and mix.
Mix in the salt, cinnamon, nutmeg and ground ginger.
Add in the baking powder, baking soda and mix until combined.
Add the flour and mix with a spatula until just combined. Don't over mix it, stop once you don't see anymore streaks of flour.
Scoop the cookies using a 1.5" or 2" wide cookie scoop and place on a tray or in a large container. Cover and refrigerate for 1.5-2 hours.
During this time, beat together the cream cheese, sugar and vanilla until smooth. Portion into the same amount of balls as the cookies and place in the freezer until the outside solidifies.
Preheat the oven to 350F.
Take each cookie and using your thumbs, make a well in the middle and widen and deepen it until the cookie is mostly flattened out.
Place the cream cheese in the middle and close the cookie around it, making sure it's sealed well and has a bit more of the cookie dough on the bottom side. Repeat with all and place on a large baking pan lined with parchment paper, with at least 2 inches of space between each cookie.
Bake for about 12-14 minutes, until the bottom is golden and the edges are set. They'll look underbaked but they will continue cooking from the residual heat from the pan. Hit them with some flakey salt right out of the oven and let them cool on the pan for 15 minutes before transferring to a cooling rack. They'll be puffy out of the oven, but will deflate as they cool.