Toast saffron in a 5- to 6-qt. pan over medium heat until fragrant, about 4 minutes; scrape out of pan and set aside. Add almonds and 1 tbsp. oil to pan and cook, stirring occasionally, until golden, about 4 minutes. With a slotted spoon, transfer nuts to a bowl.
Add remaining tbsp. oil to pan with chicken and increase heat to medium-high. Cook until chicken is brown on underside, about 3 minutes. Stir in mushrooms and garlic. Cook, stirring occasionally, until chicken is no longer pink, about 3 minutes. With slotted spoon, transfer to a bowl.
Add broth, lemon zest, pepper, saffron, and potatoes to pan. Cover and bring to a boil, then reduce heat and simmer 5 minutes. Return chicken mixture to pan and simmer, covered, until flavors are blended, about 5 minutes. Stir in spinach and cook until wilted, 1 to 2 minutes. Add lemon juice and salt to taste.
Ladle into bowls, drizzle with oil, and sprinkle with almonds.
Note: Nutritional analysis is per 1 1/2-cup serving.