I'm sure thsi has nothing to do with "french donuts" but a long time ago I received a recipe for "french donuts" which was a recipe using a yellow cake mix and orange juice as the liquid and made as muffins. While the muffins were still hot, they were dipped in melted butter and rolled in cinnamon and sugar.
Just to expand - a beignet is not typically round with a hole in the centre; they are typically square or circular but without holes.
In New Orleans, Cafe du Monde is famous for their beignets which are covered with powdered sugar (and woe to he who shows up wearing a black shirt; your server will gently puff the powdered sugar all over you!). The basic doughnut is just deep fried pate de choux.
In France, beignets are often stuffed with either fruit or savouries. They are particularly popular leading up to Lent, as a way to use up left-over fats. This is similar to English pancakes or Polish paczki, which are celebrated in the runup to Lent.
French crullers are only "French" to the extent that they are also made from pate de choux which gives them a light, fluffy texture, but I never saw one all the time I was in France.