Toast walnuts on a cookie sheet in a 350-degree oven for 7 minutes. Remove from oven, slide walnuts into a bowl and set bowl aside. Butter the cookie sheet, set it aside and turn oven up to 375 degrees.
In a pot, boil water and add 1/2 bag of turnip greens. (Save rest of greens for another use.) Lightly boil greens for about 1 minute, then drain, squeeze out excess water and set greens aside. Heat olive oil in a frying pan and saute onions and garlic until lightly browned. Add balsamic vinegar and boil to allow it to reduce slightly, about 1 minute. Add drained turnip greens and stir and cook until wilted down. Remove from heat and allow to cool slightly while you prepare the phyllo dough.
Remove and open one plastic package of phyllo, which should have 10 paper-thin sheets of phyllo in it. (There are usually 2 plastic packages in each box. Save the other package for another use.) Lay phyllo on a large cutting board. It should be a stack of phyllo sheets about 9 inches wide by 14 inches long. Cut the stack lengthwise, so you have 2 10-sheet stacks of 4 1/2- by 14-inch phyllo sheets.
To make your first Turnip Triangle, remove 3 sheets from one of your stacks, keep them together and place them on the board in front of you. Brush the top lightly with melted butter. Stir walnuts and feta into the cooked turnip mixture. Place 1/7th of the greens mixture in the corner of this buttered stack of phyllo sheets, and fold it in a 45-degree angle towards the opposite corner, much as you would if you were folding a flag for storage. Continue folding, retaining your triangle shape and brushing the packet with butter as you go.
Place filled triangle on the prepared cookie sheet and brush top with melted butter. Continue making trianges, making 7 in all. (Note: due to an uneven number, the last triangle will be made with only 2 phyllo sheets in a stack.)
Place on prepared baking sheet, and bake for 15-20 minutes until puffed and golden brown.