Preheat oven to 350 degrees F (175 degrees C). Grease a muffin tin with 1 teaspoon coconut oil.
Bring coconut milk, quinoa, and pinch of salt to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until quinoa is tender and milk is absorbed, 20 to 25 minutes. Remove from heat and cool.
Combine 2 tablespoons coconut oil and honey in a large microwave-safe bowl; heat in microwave until melted and smooth, 20 to 30 seconds. Beat eggs, shredded coconut, and vanilla extract into honey mixture until well combined. Stir cooled quinoa into mixture until well mixed.
Place dates, chia seeds, walnuts, and ground flax in a food processor; pulse until roughly chopped. Stir into quinoa batter.
Place carrot and apple in food processor and pulse until evenly chopped; stir into quinoa batter. Sprinkle a pinch of salt over batter and stir well. Divide batter into the muffin cups, filling almost full.
Bake in the preheated oven until the tops are golden brown and puffed and a toothpick inserted in the middle comes out clean, 12 to 15 minutes.