In a bowl, whisk together flour, 1 tablespoon sugar, and 1/4 teaspoon salt. Add the chilled shortening cubes and 2 tablespoons cold butter, then use a pastry cutter or your fingertips to work them into the flour until the mixture resembles coarse breadcrumbs with some pea-sized pieces remaining. Stir in the cold water until just combined into a shaggy dough. Turn onto a floured surface, knead gently 2-3 times to bring it together, then wrap and refrigerate for 30 minutes while you prepare the other components.
While the dough chills, melt 1 tablespoon butter in a saucepan over medium heat. Add the diced peaches, both sugars (granulated and brown), cinnamon, nutmeg, and lemon juice, stirring to combine. Cook for 8-10 minutes, stirring occasionally, until the peaches soften and release their juices to form a syrup. I like to leave some peach pieces chunky for texture—mash about half the mixture with a spoon while leaving the other half in pieces. Let the filling cool completely to room temperature before using.
Preheat your oven to 350°F. On a floured surface, roll the chilled dough from Step 1 to about 1/8-inch thickness. Brush with melted butter, sprinkle generously with cinnamon sugar, fold the dough in half, and repeat the brush-sprinkle-fold process two more times to create a laminated, flaky texture. Cut the dough into 1-inch squares and spread on a baking sheet. Bake for 15-20 minutes until golden and crispy. Once cooled, break or crush the squares into bite-sized pieces—you want a mix of sizes for texture.
In a large bowl, whip the heavy cream with an electric mixer until stiff peaks form—this will take about 3-4 minutes. Gently fold in the sweetened condensed milk, 1.5 tablespoons vanilla extract, 1/2 teaspoon cinnamon, and a pinch of salt until just combined. I find that folding rather than stirring keeps the cream light and airy, which gives you that premium ice cream texture. Gently fold in the cooled peach filling from Step 2 and about 1 cup of the crushed cobbler pieces from Step 3.
Pour the ice cream mixture into a 9x13-inch baking dish or similar-sized container. Smooth the top, then scatter the remaining crushed cobbler pieces over the surface, pressing them in slightly so they stay distributed throughout as it freezes. Cover with plastic wrap or foil and freeze for at least 4 hours, though overnight is even better for scoopability. You'll have a delicious dessert that captures all the flavors of a warm peach cobbler in creamy, frozen form.