Rinse buckwheat with cold water in a colander; drain.
Heat a skillet over medium heat; cook and stir buckwheat until toasted and fragrant, 5 to 10 minutes. Transfer to a bowl.
Heat olive oil in the skillet over medium heat; cook and stir onion and carrot until onion is golden, 5 to 10 minutes. Add mushrooms; cook and stir for 5 minutes more.
Melt butter in a pot over medium heat; add buckwheat and stir to coat. Add onion mixture, water, salt, and pepper; bring to a boil. Reduce heat, cover pot, and simmer until liquid is absorbed, about 20 minutes.