Heat a large frying pan over medium-high heat; add olive oil. Add onion and cook, stirring, until translucent, about 3 minutes. Add garlic and cook 2 minutes more.
Add Swiss chard and dandelion greens to pan in large handfuls and cook, stirring, until any liquid is evaporated. Transfer mixture to a medium bowl and stir in parmesan, ricotta, egg, salt, and pepper.
Preheat oven to 350°. Melt butter in a small saucepan. Unroll phyllo and cover with a moist towel. Working with 1 sheet of phyllo at a time, arrange sheet so a short side is facing you. Cut sheet lengthwise into three even strips and brush each strip liberally with butter. Place a heaping tbsp. of greens at the bottom of each strip of phyllo, slightly off center. Fold the dough to make a triangle, then continue folding and turning (like you are folding a flag) to make the turnover. Brush top with butter and put on a baking sheet. Repeat with remaining phyllo, greens, and butter. Bake finished turnovers until puffed and brown, about 30 minutes. Serve hot.
Note: Nutritional analysis is per turnover.