Preheat oven to 400° and line a large baking sheet with parchment. In a large bowl, toss peaches, butter, granulated sugar, lemon juice, cinnamon, ginger, nutmeg, and salt to coat. Scrape peach mixture onto prepared sheet, arranging in a single layer.

Roast peaches just until tender and juicy, 10 to 12 minutes. (Don’t overbake or the fruit will be too soft and mushy.) Let cool. Transfer half of peaches to an airtight container and refrigerate until ready to use. (The remaining half will be layered in the middle of the cheesecake.)

Reduce oven temperature to 350° and line another large baking sheet with parchment. In a medium bowl, mix flour, brown sugar, cinnamon, and salt. Add butter and stir until mixture resembles coarse crumbs.

Dump onto prepared sheet and spread in a single layer. Bake crumble until lightly golden brown, 8 to 10 minutes. Let cool and refrigerate until ready to use.

Arrange racks in center and lower third of oven; preheat to 350°. Grease a 9" springform with cooking spray.

In a large bowl, mix graham cracker crumbs, butter, granulated sugar, cinnamon, and salt until combined (mixture should resemble wet sand). Press into bottom and about 1" up sides of prepared pan.

Bake crust on center rack until fragrant and dry to the touch, 8 to 10 minutes. Transfer to a wire rack and let cool.

Reduce oven temperature to 325°. Fill a roasting pan or large baking dish with water and place on lower oven rack. In a large bowl, using a handheld mixer on medium-high speed (or in the large bowl of a stand mixer fitted with the whisk attachment), beat cream cheese and granulated sugar on medium-high speed until smooth, about 1 minute. Reduce mixer speed to medium-low speed and add eggs, one at a time, beating after each addition. Fold in sour cream and vanilla. Add flour and salt and beat until just combined.

Pour half of batter into cooled crust, then top with room-temperature roasted peaches in an even layer. Sprinkle with one-third of crumble. Top with remaining batter and smooth top.

Bake cheesecake on center rack until center only slightly jiggles, 1 hour 15 minutes to 1 hour 30 minutes.

Turn off oven, prop open door, and let cheesecake cool in oven, about 1 hour. Remove cheesecake from oven and let cool to room temperature. Refrigerate until chilled, at least 6 hours or up to overnight.

When ready to serve, remove sides of pan and arrange reserved refrigerated peaches on top. Top with remaining crumble.