Preheat oven to 375°F. Line a sheet tray with parchment or a Silpat and lightly spray with oil.
In a large bowl, combine sweet potatoes, broccoli slaw, and salt. Let sit 5–10 minutes to release moisture.
Pat dry, then transfer to a bowl. Add tapioca starch, pepper, garlic powder, and 1 egg. Mix well.
The mixture should be slightly dry but able to hold together. Add more tapioca starch if needed.
Divide into 6 mounds on the tray, shaping each into a “nest” with a hollow center. Lightly spray tops.
Bake for 15–20 minutes, until tender.
Remove and brush with pesto (optional).
Crack an egg into each nest.
Return to oven and bake 5–15 minutes, depending on desired egg doneness.
Drizzle with olive oil and serve.