In a 3-quart saucepan cook onion and garlic in hot oil over medium heat for 3 to 5 minutes or until onion is tender, stirring occasionally. Add the rice; cook for 3 to 5 minutes or until rice is golden brown, stirring frequently.
Meanwhile, in a 1 1/2-quart saucepan bring broth to boiling; reduce heat. Cover and keep broth simmering. Carefully stir 1/2 cup of the broth into the rice mixture. Cook over medium heat until liquid is absorbed, stirring frequently. Continue adding broth, 1/2 cup at a time, stirring frequently until the broth has been absorbed.
With the last addition of broth, add peas and carrot. Rice should be tender and creamy. (This should take 20 to 25 minutes total.)
Stir in cheese, spinach, and snipped fresh thyme with the cheese. (If you prefer to use dried thyme, add 1/2 teaspoon dried thyme, crushed, with the carrot.) Heat through. Serve immediately.