Bring a medium 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 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. Portion out half for the dog and let cool before serving; season the people's portion with salt and pepper.