Preheat oven to 375°F (190°C). Grease a 9-inch springform pan and line the base with parchment paper.
Whisk together flour, 1/4 cup sugar, baking powder, and salt. Cut in cold butter until crumbly. Add egg and vanilla, mixing just until dough comes together.
Press dough evenly into the pan, bringing it up the sides about 1.5 inches high. Prick with a fork and chill while preparing peaches.
Peel and slice fresh peaches, or drain canned peaches thoroughly and pat dry. Toss slices with 2 tbsp sugar and 1/2 tsp cinnamon.
Arrange peach slices over the chilled crust in a single overlapping layer, starting from the edge and working inward.
Whisk eggs, sour cream, 1/2 cup sugar, vanilla, 1/4 tsp cinnamon, and a pinch of salt until completely smooth.
Pour custard slowly over the peaches, gently shaking the pan to settle it evenly between the fruit.
Bake for 50 to 60 minutes, until custard is set and no longer jiggles in the center. Tent with foil if browning too quickly.
Cool in pan for 20 minutes on a rack. Release the springform, dust with powdered sugar, and serve warm or at room temperature.