How to Make It
In a large skillet, heat 3 tablespoons of the oil. Add the garlic and shallots and cook over moderately high heat, stirring occasionally, until softened. Add the goat and cook, stirring frequently, until lightly browned, about 8 minutes. Tilt the pan and spoon off as much fat as possible. Add the whole Thai chiles, 1/4 cup of the fish sauce, the palm sugar and 1/2 cup of water to the skillet and simmer until the sugar is dissolved. Stir in the basil and keep warm.
In a small bowl, combine the sliced chiles with the remaining 1/4 cup of fish sauce.
In a large nonstick skillet, heat the remaining 1/4 cup of oil. Fry the eggs, 2 at a time, over high heat, tilting the pan and spooning the hot oil over the eggs to puff them up.
Spoon the stir-fried goat over steamed rice and top with the fried eggs. Serve the chile-fish sauce on the side.