Preheat oven to 375°F (190°C). Line a 9-inch square baking pan with heavy-duty foil, letting excess extend over sides of pan. Lightly spray foil with cooking spray.

In a large bowl, whisk together flour and baking powder.

In a medium microwave-safe bowl, heat 1 cup (227 grams) butter on high in 10-second intervals, stirring between each, until melted; whisk in 1 cup (200 grams) granulated sugar, 1½ teaspoons (4.5 grams) salt, and 1 teaspoon (4 grams) vanilla. Stir butter mixture into flour mixture until combined. Reserve ¾ cup (217 grams) dough in a small bowl; crumble. Press remaining dough into bottom of prepared pan.

Bake until edges are lightly browned, 10 to 12 minutes. Let cool slightly in pan on a wire rack. Leave oven on.

Meanwhile, in a medium saucepan, whisk together cornstarch, cardamom, remaining ⅔ cup (133 grams) sugar, and remaining ½ teaspoon (1.5 grams) salt. Stir in 3 cups (480 grams) peaches, lemon juice, and remaining 3 tablespoons (42 grams) butter until well combined. Cook over medium heat, stirring frequently, until bubbles begin to form and liquid is thickened, 5 to 6 minutes.

Remove from heat; stir in remaining 2 cups (320 grams) peaches and remaining ½ teaspoon (2 grams) vanilla. Spread peach mixture evenly onto prepared crust, pressing thickened liquid between peach pieces. Sprinkle reserved dough onto peaches.

Bake until top is golden brown and filling is bubbly, about 25 minutes, loosely covering with foil to prevent excess browning, if necessary. Let cool completely in pan on a wire rack.

Using excess foil as handles, remove from pan, and cut into bars. Store in an airtight container for up to 3 days.