Heat the heavy cream until warm in the microwave or in a small saucepan. Set aside. In a heavy-bottomed medium, placed over moderately high heat, pour in the sugar. Cook until the sugar begins to melt and turn a light golden brown around the edges. Using a silicon spatula, begin to stir until the sugar melts completely and turns a lightly golden brown. Cook for an additional 30 seconds or so until the sugar turns a golden brown color. Immediately take the pan off the heat and stir in the butter. Once the butter has melted, pour in the warm heavy cream. The mixture will bubble up, not to worry - this is normal, just keep stirring until smooth. If the mixture seizes, place the caramel over very low heat and keep stirring until it becomes smooth. Set aside.
In a medium bowl, mix the cubed unsalted butter, rolled oats, flour, sugar, cinnamon and salt. Break up the cubed butter until it resembles the size of peas and evenly distributed throughout. Transfer to the refrigerator until you're ready to use.
Roll out your pie dough and fit in it in a 9-inch pan. Crimp the edges, if you like. Transfer to the freezer until we're ready to use fill it.
Meanwhile, preheat the oven to 425 degrees F. In a large bowl, juice the 1/2 lemon and toss with the pear slices and white sugar; allow to sit and macerate, about 15 minutes. To the bowl, add the light brown sugar, all-purpose flour, cinnamon, ginger, cardamom, salt and vanilla extract; toss until the pears are evenly coated.
Remove the pie crust from the freezer (it should've been in there now for about 30 minutes and your oven should be nice and hot). Fill the pie crust with the pear mixture, pressing them down so the mixture is tightly packed. Pour the caramel over the pear mixture. Note: If the caramel has stiffened a bit, you can warm it up in the microwave for 30 seconds or so; you just want it pourable--it'll thin out in the oven. Top the caramel-pears with the oat crumble. Brush the crimped edges with egg wash.
Transfer to the oven for 20-25 minutes. At the 20 minute-mark, give it a check. If the edges are golden brown, remove from the oven and cover the edges with foil (I had to do this.). Bring the heat down to 325 degrees F and bake for an additional 25 minutes until the pears are tender when poked with a knife and caramel has bubbled up. Remove and allow to cool for 2 to 3 hours.