Make the compote:
Combine cranberries, brown sugar, and orange zest and juice in a small, heavy-bottomed saucepan. Cook over medium-low, stirring occasionally, until cranberries begin to burst, and mixture thickens slightly, 14 to 18 minutes. Set aside to cool, about 1 hour.
Make the crumble:
Stir together oats, flour, brown sugar, cinnamon, and salt in a large bowl. Add butter, and toss to coat in oat mixture. Using a bench scraper, cut butter into 1/2-inch cubes. Using your hands, quickly separate butter cubes, and toss to coat. Using bench scraper, cut butter pieces in half to form 1/4-inch pieces.
Using a pastry blender, cut in butter, rotating bowl after each cut, until butter is mostly incorporated, stopping to clean pastry blender as needed. Using your fingers, smear any large butter pieces to fully incorporate until butter is no longer visible, being careful not to overwork mixture.
Make the filling:
Preheat oven to 350°F. Line a baking sheet with parchment paper. Stir together sugar, tapioca starch, salt, cinnamon, nutmeg, and allspice in a large bowl. Add cranberries, pear, and cooled compote. Mix until well-combined.
Assemble the pie:
Using a small offset spatula, evenly spread cream cheese on bottom of blind-baked pie shell. Brush crimped edge with beaten egg. Spoon cranberry mixture on cream cheese, and spread in an even layer. Pile crumble over filling, leaving a small (about 1 1/2-inch) hole in center of pie to serve both as a steam vent for the fruit as it cooks and as an indicator of when the pie is done. Place assembled pie on prepared baking sheet.
Bake in preheated oven until juices are just bubbling in center and topping is golden brown, 1 hour and 5 minutes to 1 hour and 15 minutes. Transfer pie to a wire rack to cool completely, 4 to 6 hours. Serve with ice cream.
Compote and crumble can be refrigerated in airtight containers up to 4 days. Store leftover pie, covered, at room temperature up to 2 days.