First, preheat your oven to 375°F (190°C). Lightly grease and flour a 9x13 inch baking pan, or line it with parchment paper, leaving some overhang on the sides.
Prepare your peach filling. If you're using fresh peaches, slice them into bite-sized pieces. If using frozen, make sure they are completely thawed and drain off any excess liquid really, really well. In a medium bowl, gently toss the peaches with the brown sugar and cinnamon. Set this aside while you make the crumble.
Now, let's make the crumble mixture! In a large bowl, whisk together the flour, granulated sugar, and salt. Add the cold, cubed butter and vanilla extract.
Work the butter into the dry ingredients. You can use a pastry blender, a fork, or even your clean fingertips. Cut or rub the butter into the flour mixture until it resembles coarse crumbs, with some pea-sized pieces of butter remaining.
Take about 3/4 of this crumble mixture and place it back into the large bowl. Add the heavy cream and mix until just combined. It will be a bit crumbly but should hold together when you press it.
Press this cream-enhanced mixture evenly into the bottom of your prepared baking pan. Press it down gently but firmly to form the base layer for your bars.
Carefully spoon the peach filling evenly over the crust layer. Try to leave behind any excess liquid that might have accumulated in the bottom of the peach bowl.
Take the remaining 1/4 of the original dry crumble mixture (the part you didn't add cream to) and sprinkle it evenly over the peach layer. This will be your lovely, crisp topping!
Bake for 40-50 minutes, or until the topping is golden brown and the peach filling is bubbly around the edges.
Let them cool COMPLETELY in the pan on a wire rack before cutting. Cooling allows the bars to set properly so they don't fall apart. Wait at least 2-3 hours, or even pop them in the fridge for an hour once they've cooled down a bit.
Once fully cool, use the parchment paper overhang to lift the whole slab out of the pan. Slice into squares or rectangles with a sharp knife.