Preheat the oven to 180°C. Peel and slice the beets. Coat a 24 cm flan dish with olive oil and cover the bottom with beet slices; halve the rest and line the edges, round sides up. Brush with olive oil, season with salt and pepper, and bake the crust for 15 minutes.
Mince the shallots. Heat a pan over medium heat with olive oil and add the shallots; when golden, add the spinach and sauté for 3 minutes. Remove from heat and set aside.
Spread the shallot and spinach mix over the beetroot crust. Add the goat cheese and smoked chicken.
In a bowl, whisk the eggs and cream with salt and pepper to taste. Pour over the filling.
Cover with cheese slices and bake for 30–35 minutes, until set. Serve with rocket leaves.