In a large saucepan, over medium high heat, bring to a boil the broth, water, 1 of the minced garlic cloves, carrots, turnips and half of the saffron. Cover and simmer until the vegetables are tender but still crisp, about 5 minutes. Meanwhile cut the chicken breasts into 3/4-inch dice. Finely chop the roasted red peppers and combine them with remaining minced garlic clove, remaining saffron and mayonnaise. Season to taste with salt and 1/8th teaspoon cayenne.
When the vegetables are tender, add the chicken, bring the broth back to a simmer, and cook, uncovered, for 2 minutes or until the chicken is just cooked through. Season to taste with salt and cayenne.
Spread each toast with saffron mayonnaise and set each one in a soup bowl. Ladle the steaming broth, vegetables and chicken over the bread and serve immediately. Leftover ideas: If you have half of the soup and the mayonnaise leftover the next day, then this is what you do. Bring the soup to a boil. Whisk one egg into remaining mayonnaise and ladle some hot soup into the mayonnaise to temper it. Whisk the egg mayonnaise back into the hot soup and stir, over low heat until broth has thickened. Do not boil the broth or the egg will curdle Remove from heat, adjust seasoning and serve right away.