Fill a medium pot with water, about halfway full. Bring water to a boil, as you prepare the turnips:
Use a vegetable peeler or sharp knife to peel the turnips. Cut each turnip in half, as shown.
Place a turnip half - cut side down - on a cutting board, with a chopstick along each long side.
Make vertical cuts from the top curve of the turnip, down to where the chopsticks are. Be careful to keep each cut straight up and down, and relatively close together.
Once you have the entire turnip half covered with these cuts, repeat with remaining halves,
Place turnips in the boiling water, boil for 20 minutes.
As turnips are boiling, preheat oven to 375 F. Line a baking sheet with parchment paper, set aside.
Once the boiling time is over, use a slotted spoon to carefully remove cooked turnips from the water. Blot with paper towels to remove excess moisture.
Arrange cooked turnips on the prepared baking pan, flat cut side down.
In a microwave safe small bowl, melt butter. Add garlic, parsley, and salt, mix well to combine.
Use a pastry brush to brush most of the melted butter over the turnips.
Bake in the preheated oven for 30 minutes, brushing them again with remaining butter - and butter that has pooled on the pan after cooking - after about 20 minutes.
If you’d like, you can broil them on high for the last few minutes, to crisp them up a bit.
For cheesy hasselback turnips, use a cheese slicer or sharp knife to cut a pile of really thin cheese slices.
Once the 30 minute baking time is over, remove pan from the oven. Carefully insert cheese slices (breaking to fit) in each of the cuts. Return pan to the oven and broil on high for a few minutes to melt and brown the cheese.
Serve hot!