Heat 1 tbsp olive oil in a large skillet over medium-high heat. Add mushrooms and garlic. Cook until mushrooms are tender, approximately 3 to 4 minutes, stirring often.
Add spinach to the mushroom mixture. Cook until spinach is wilted, approximately 2 to 3 minutes, stirring occasionally.
Remove from heat and add spinach mixture to bowl of food processor. Add remaining 1 tbsp olive oil, lemon juice, and lemon zest. Pulse to combine. Set aside until completely cooled.
In a large bowl, combine chicken, salt, and tapioca flour.
To the chicken mixture, add cooled spinach mixture and coconut flour. Mix well to combine. Let stand for 5 minutes. Mixture should be wet and no pooling liquids in bowl.
Divide mixture into 5 equal portions and shape each portion into patties.
Over medium heat in the skillet used to cook mushrooms and garlic, heat just enough olive oil to lightly cover bottom of skillet.
Carefully place patties in skillet. To avoid overcrowding, you might have to cook the patties in batches.
Cook, flipping with a spatula, until patties are golden brown and cooked through with an internal temperature of 165 degrees F, approximately 5 minutes per side.
Remove from heat and serve with your favorite side dish.