Heat a large heavy-bottom or non-stick skillet over medium high heat. Add the oil.
Add the cubed butternut squash to the hot pan and toss in the oil.
Let the butternut squash cook for about 5 minutes, tossing a few times gently. Add the maple syrup and the brown sugar, along with the garlic and salt and pepper. Turn the heat down to medium.
Toss the squash in the pan again until it's coated in the syrup mixture and let it cook as it lightly browns on all sides, turning the pieces gently every few minutes until they're just barely fork tender.
Add the chopped beet greens and stir them together with the squash. Add a lid to the pan and set the heat to low. Let the greens steam for about 3 minutes.
Add the squash and greens to a platter and add the crumbled goat cheese over the top.