Grate the butternut squash, using the medium holes of the grater.
In a medium pan large enough to spread the squash in one layer, heat 1 ½ tablespoons of olive oil, once it starts to shimmer, add the grated butternut squash spread it out so it is in one thin layer and cook for about 2-3 minutes. Let it brown then mix it a bit so all parts have browned for about 2 more minutes.
Remove from the pan and place in a plate.
Brush the bread with olive oil on both sides.
Smash the feta in a bowl. Spread half of the feta on one slice of bread and the other half on the other slice of bread. Layer the cooked butternut squash on one slice and cover with the other slice .
Grill the sandwich in the pan that you used to cook the butternut squash. Grill about 2-3 minutes on each side, depending on the thickness of the bread.