Line a 9-inch square metal baking pan with parchment, leaving overhang on two sides. Heat the oven to 340°F (171°C).
Spread the oats on a sheet pan and toast for 6–8 minutes, stirring once, until fragrant and lightly golden. Cool completely.
In a bowl, toss the sliced peaches with 1 tbsp sugar, lemon zest, pinch of cinnamon, and 2 tsp flour. Set aside and drain off any juices before assembling.
Make the cheesecake filling: Beat cream cheese, granulated sugar, brown sugar, cinnamon, and vanilla until smooth and creamy, 1–2 minutes. Beat in the egg on low just until combined. Refrigerate the filling while you prepare the crust.
For the crust, whisk together the melted butter, brown sugar, granulated sugar, egg, and vanilla in a large bowl until glossy. In a separate bowl, whisk the flour, cinnamon, baking soda, and salt. Add the dry ingredients to the wet and stir just until evenly moistened; fold in the cooled toasted oats. The dough will be thick.
Press about two-thirds of the dough firmly and evenly into the prepared pan. Dock lightly with a fork, freeze for 10 minutes, then par-bake for 8–10 minutes until the edges look set.
Cool the base on a rack for 5 minutes. Lightly blot peaches with a paper towel, then arrange them snugly in an even layer over the warm crust.
Stir the cheesecake filling and spread it gently over the peaches, smoothing to the corners without disturbing the fruit.
Pinch the remaining dough into small clumps and scatter over the top so some cheesecake peeks through.
Bake for 24–28 minutes, or until the crumble is golden and the center wobbles slightly when the pan is nudged. Do not overbake; it will set as it cools.
Cool on a rack for 1 hour, then chill at least 3 hours (or overnight). Lift out with the parchment, and slice into 16 bars. For ultra-clean cuts, warm a knife under hot water, wipe dry, and cut.