A food processor is the easiest way to process the carrots but you could cut everything up by hand. It will definitely add to the prep time. Process the carrots using the on/off or pulse button on your processor until the carrots are finely cut up.
Heat your saucepan over medium heat, add butter. When the butter is melted, add the orzo and carrots and cook (saute) until the orzo is golden. This should take about 5 minutes.
Add the water, chicken stock and minced garlic. Let this all cook uncovered until all the water and chicken stock are absorbed into the orzo being sure to stir frequently. This should take about 10 minutes.
Now it's time to add the Parmesan cheese, green onions and rosemary. Taste and adjust seasonings with salt & pepper.