Step 1

For the Crumble: In a medium heatproof bowl, combine all-purpose flour, salt, cinnamon, and five-spice powder.

Step 2

In a blender, quickly pulse almonds until roughly chopped, about 5 seconds. Add to flour mixture.

Step 3

In the same blender, combine granulated sugar, brown sugar, and thyme. Process until thyme is finely chopped and evenly incorporated into sugar (the mixture should resemble wet sand), about 30 seconds. Add to bowl with flour mixture, then whisk together until well combined, about 30 seconds.

Step 4

Pour melted butter over flour mixture. Using a flexible spatula, thoroughly mix until no dry flour remains and a damp, crumbly meal has formed, about 1 minute. Set aside.

Step 5

For the Filling: In a 1-gallon zipper-lock bag, combine brown sugar, salt, cinnamon, five-spice powder, and white pepper.

Step 6

Peel, quarter, and core apples. Slice each quarter lengthwise into 1/2-inch thick wedges (you should have about 20 ounces/565g of sliced apples, measuring about 5 rounded cups). Add sliced apples to bag, seal, and shake until coated. Macerate at room temperature, flipping the bag occasionally to distribute the syrup, until apples have lost a third of their volume, at least 3 hours. The apples can be held up to 8 hours in the refrigerator.

Step 7

Meanwhile, peel then quarter sweet potatoes lengthwise. Slice each quarter crosswise into 1/2-inch thick wedges (you should have about 20 ounces/565g of sliced sweet potatoes, measuring about 4 rounded cups).

Step 8

Place sweet potatoes in a 3-quart saucepan and cover with cold water. Bring to a boil over high heat, then reduce heat to low to maintain a simmer. Cook until sweet potatoes are tender enough to be pierced with a knife, but not completely cooked through, about 5 minutes. Drain sweet potatoes and run under cold water until cooled to stop cooking. Set aside to let cool to room temperature, about 15 minutes.

Step 9

Adjust oven rack to lower-middle position and preheat to 400°F (204°C).

Step 10

When apples are ready, add sweet potatoes and tapioca starch to bag, reseal, and toss to combine. Pour sweet potato and apples, along with accumulated syrup, into the prepared pie shell, gently pressing the slices into a flattish mound.

Step 11

Evenly distribute crumble topping over pie filling, breaking up any large chunks with your fingers.

Step 12

Place pie on a parchment-lined half-sheet tray. Bake to an internal temperature of around 195°F (91°C) in the very center, about 1 hour 15 minutes (if crumble topping starts to darken beyond chestnut brown in color, loosely cover pie with aluminum foil for remainder of baking time).

Step 13

Transfer pie to a wire rack and let cool to room temperature, about 1 hour. Slice and serve.