Bring a medium size pot of water to a boil, salt it, add the orzo and cook until just before al dente, about 7 minutes. Add the peas and cook for 1 minute; drain and return the pasta and peas to the pot.
While the pasta is cooking, in a medium size saucepan, cover the carrots with water and bring to a boil. Season with salt and cook until tender, about 12 minutes. Drain, reserving about 1/2 cup cooking water, then transfer the carrots to a food processor. Add the reserved cooking water and honey, if using, and process until smooth.
Stir the carrot puree into the pasta and peas. Stir in the Parmigiano Reggiano cheese. Portion out half for the dog and let cool before serving; season the people’s portion with salt and pepper.