In a medium saucepan, heat 1 tablespoon of the olive oil over medium low heat. Add the garlic and cook, stirring frequently, until golden. Do not allow it to burn. Add the chicken stock, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Increase the heat to medium high and bring the mixture to a simmer. Add the mint leaves and remove the pan from the heat. Cover the pan and let the mixture steep for 30 minutes, stirring 2 or 3 times.
Meanwhile, preheat the oven to 375 degrees. Place the bread cubes in a bowl and pour the remaining 3 tablespoons of oil over them. Add a pinch each of salt and pepper and toss well. Spread the bread out on a baking sheet and bake for 10 to 15 minutes, or until golden and crisp. Strain the mint-infused mixture into a clean saucepan and bring the mixture to a boil over medium high heat. Drizzle the eggs over the boiling liquid and stir gently for 1 minute to cook them. Place some of the toasted croutons in the bottom of each of 6 soup bowls, preferably earthenware, and ladle the hot soup over the top. Garnish each bowl with a sprig of mint and a wedge of lime and serve.