Bring water to a boil in a medium saucepan. Add rice, cover and cook over low heat until barely tender (about 15 minutes).
Meanwhile, in a medium saucepan, bring the stock to a simmer over moderately high heat. Add the chicken and simmer until cooked through (about 10 minutes). Using a slotted spoon, transfer the chicken to a plate. Skim any fat from the surface of the stock.
In a large saucepan, whisk together yogurt, egg yolk and flour. Whisk in stock and bring to a simmer over low heat, whisking occasionally. Stir in the chickpeas, rice and reserved chicken. Season with salt and pepper and keep warm over low heat.
Melt butter in a small skillet. Add the garlic, paprika and cayenne and cook over medium heat until fragrant (about 2 minutes). Ladle the soup into bowls and swirl in the garlic butter. Sprinkle with the mint and serve with lemon wedges.