Whisk the apple cider vinegar, salt, and sugar together until the sugar and salt have dissolved. (If you are using turbinado sugar, I suggest you heat this mixture for about 30 seconds to help the sugar dissolve.)
Peel the cucumbers and cut them lengthwise into strips. Using a spoon, scoop out the seeds.
Place the cucumbers in the vinegar brine, give them a stir so they are well coated, and set this aside for about 30 minutes.
Heat the oven to 375F degrees.
In a small sauté pan, melt the butter and sauté the onions for a few minutes until they are transparent.
Add the chopped basil and pepper and stir to combine.
Pour the butter and onion mixture into a 7" x 9" casserole dish and use this mixture to coat the bottom and sides of the dish.
Drain the cucumbers and pat them dry with a paper or cloth towel.
Place the cucumbers in the butter mixture and stir to coat the cucumbers with the butter.
Bake the cucumbers in a 375F degree oven for about one hour, stirring them once or twice during that time.
About 5-10 minutes before the cucumbers are ready to remove from the oven, sprinkle the top with the Pecorino Romano cheese.
The cucumbers will still be a little crisp and only a tiny bit browned when they are done.
Serve hot.
Notes
Note: Marinating the cucumbers in a vinegar/salt brine helps to remove some of the moisture from the cucumbers. It also removes any bitterness that may occur if your cucumbers are over ripe as mine were.