Bring a large pot of lightly salted water to a boil. Cook orzo in the boiling water, stirring occasionally until cooked through but firm to the bite, about 11 minutes. Drain, reserving some of the water.
Combine spinach and beets in a blender adding as much of the reserved water as needed to help the mixture blend smoothly; blend until smooth.
Melt butter in a large pot over medium heat. Stir blended spinach mixture with the melted butter; cook until the color begins to change, 5 to 7 minutes. Stir water into the mixture to thin the liquid. Stir drained orzo pasta into the soup; season with thyme, salt, and pepper. Remove pot from heat and stir sour cream through the soup. Sprinkle with Cheddar cheese to serve.