For the aioli, tear the bread into pieces and soak in the milk. Remove the bread from the milk, squeeze well, and place in a bowl. Peel the garlic and pass through a press into the bread mixture. Add the yolks, and stir until thoroughly combined. Gradually incorporate the oil until a thick sauce has formed. Season to taste with chili powder, ketchup, salt, pepper, and lemon juice.
Thinly slice the chili, for garnish. Place the aioli in small bowls, garnish with the basil and chili, and serve.