Place the peeled and quartered peaches into the bowl of a food processor. Pulse in short bursts until the fruit is finely chopped but not puréed. Set aside.

In a medium bowl, sift together the flour, baking powder, baking soda, cinnamon, and salt. Set aside.

In a large mixing bowl, combine the brown sugar and canola oil. Stir well until any lumps in the sugar have dissolved. Incorporate the chopped peaches and heavy cream. Mix until fully combined, then add the eggs one at a time, stirring gently between each addition.

Gradually fold the dry mixture into the wet mixture in two batches. Mix only until the batter is just combined—do not overmix. Cover and refrigerate the batter for 30 minutes to allow it to firm slightly.

Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper. Using a tablespoon or a small cookie scoop, drop heaping spoonfuls of the chilled dough onto the prepared sheets, spacing them approximately two inches apart. Bake for 10 to 12 minutes or until the tops are firm and a toothpick inserted into the center comes out clean. Allow the cookies to cool on the baking sheets for 10 minutes before transferring them to a wire rack to cool completely.

In the bowl of a stand mixer fitted with the paddle attachment, beat the softened cream cheese and powdered sugar until smooth and fluffy. Add the vanilla extract, peach schnapps, and finely chopped peaches. Continue beating until the mixture is well incorporated and smooth.

Turn half of the cooled cookies flat-side up. Using a spoon or piping bag, place a generous dollop of the peach cream cheese filling in the center of each. Place another cookie, flat-side down, over the filling and press gently until the filling spreads evenly to the edges. Repeat until all cookies are paired and filled.