Heat oil in a saucepan over medium heat; add onion and season with a pinch of salt. Cook and stir onion until translucent, about 5 minutes. Stir flour with the onion; cook, stirring continually, for 1 minute.

Pour 1 cup milk into the saucepan while stirring; add goat cheese and stir to melt. Stir remaining milk into the mixture, 1/2 cup at a time, and cook until you reach your desired texture, 5 to 10 minutes more; season with 1/2 teaspoon salt and black pepper.