In a large bowl, stir together the milk and yeast. Stir in the sugar, eggs and vanilla and mix well. Add a cup of the flour and the salt, then gradually add the rest of the flour, stirring and then kneading for a few (or several) minutes, until it's smooth and elastic, and still a little tacky. Transfer to a baking sheet and cover with plastic wrap; chill in the refrigerator for 30 minutes.
Meanwhile, beat the butter and flour with an electric mixer for a couple minutes, scraping down the sides of the bowl, until smooth.
When the dough has chilled, turn it out onto a lightly floured surface and roll into a rectangle that is about 13"x18" and 1/4" thick. Spread the butter evenly over the dough, then fold it as you would fold a letter, in thirds. (Unlike a letter, the dough ends should line up, so that it's folded in three.) Cover the dough in plastic wrap and put it back into the fridge for 30 minutes.
Pull the dough out and put it back on the countertop, with the open sides to the left and right. Roll it out into another 13"x18" rectangle, 1/4" thick. Fold the left third over the middle, then the right third over the middle. (This is referred to as "turns". To keep track of each fold -or turn- press your finger into the dough at the edge to make two marks - you can do this each time you roll and fold so that you know how many times you've done it.) Chill the dough for another 30 minutes.
Roll, fold and refrigerate the dough two more times, so that you've done it four times total. Cover and refrigerate for at an hour, or overnight.
In a heavy pot (or deep fryer) heat a couple inches of oil to about 350F, or until it's hot but not smoking, and a scrap of bread sizzles when you dip it in. Cook the doughnuts in batches, without crowding the pot (which can cool down the oil), flipping as necessary until deep golden. Transfer to a baking sheet lined with paper towel.
Meanwhile, whisk together the icing sugar, maple syrup and enough milk, water or cream to make a drizzling consistency. Drizzle over the croissoughnuts while they're still warm.