Put the red lentils with water and chicken stock in a medium pot. Sprinkle some salt and pepper to taste.
Half-close the lid and let it boil. When it boils, discard the foamy part on top with a small strainer or a wooden spatula. Then close the lid and reduce the temperature to simmer. When the red lentils are very soft, set aside.
Meanwhile, the red lentils are simmering; heat the olive oil and butter in a small pot. Add the flour to the pot and cook them to make the roux. When the bubbles are on top, pour 1 cup of cooked red lentils with water, then mix it. Pour this mixture into the lentils pot and cook them together.
Pulse the lentils with a hand blender until it's homogeneous and smooth, then cook for another 10 minutes.
Heat the olive oil and butter in another small pot and add the cayenne. Heat for about 30 seconds, then set aside.
Ladle the lentil soup to the serving bowl and drizzle the oil and cayenne mixture on top before serving. Serve with lemon wedges.