Place chicken in a pressure cooker. Close cooker securely and place pressure regulator over vent according to manufacturer's instructions. Heat until steam escapes in a steady flow and makes a whistling sound, 5 to 10 minutes. Adjust temperature until regulator is gently rocking. Cook for 15 minutes.

Let pressure release naturally according to manufacturer's instructions, 5 to 10 minutes. Unlock lid and remove.

Remove skin and bones from the chicken; chop meat into bite-sized chunks. Transfer to a large pot. Add carrots, half-and-half, stock, water, garlic, and bouillon cubes. Bring to a boil, stirring frequently. Add gnocchi and Parmesan cheese. Reduce heat to medium heat. Let simmer, stirring frequently, until flavors combine, about 5 minutes. Stir in spinach; cook until wilted, about 2 minutes. Remove from heat immediately.