Peel the potatoes and yam, then roughly chop. Transfer to a medium pot, cover with water and bring to a boil.
Simmer at medium low, covered, until the root vegetables are soft (approximately 20 minutes).
To prepare the filling, heat two teaspoons of coconut oil over medium heat in a large skillet.
Thinly slice the white parts of two leeks, and add them to the skillet.
Peel the carrot and garlic cloves, then pulse them together in a food processor until finely shredded (alternatively, dice very finely). Add to the leeks in the skillet, along with the ground beef. Cook until the beef is no longer pink.
Add the thyme, bay leaf, and tomato paste, stir. Remove from heat and allow to cool, then pack evenly into an 11" oval baking dish.
To finish the topping, remove the potatoes and yams from the pot, cool enough to handle, then mash the potatoes until very smooth using a food processor, with ⅓ cup coconut milk and ¼ tsp salt. Set aside.
Divide the mashed potatoes in half, then puree one half with ¼ cup parsley leaves until uniformly green.
Mash the yams with ¼ salt in a food processor . Set aside.
Put together a pastry bag, coupler and large round frosting tip. Fold the top half of the bag over to form a cuff, then spoon in the white mashed potatoes. Slowly squeeze the potatoes down into the tip, then out in a ribbon across the surface of the filling.
Leaving an empty space for the yams and green potatoes, continue piping potato out until the dish has been done end-to-end.
Clean the pastry bag, then fill with mashed yams. Repeat the process, leaving room for the green potatoes.
Clean the pastry bag, then fill with green potato and pipe out to fill the last remaining spaces across the dish's surface.
Bake in the preheated oven for 25 minutes until lightly browned and bubbly around the edges. Serve warm.