To do both, one has to have enough body fat to support the workouts - when it is burned to replace the missing calories from the restricted diet it is turned slowly into fuel instead of eating all the carbohydrates, proteins, and fats. If you decide to do this, be aware that you can end up catabolizing muscle instead of burning fat - thereby reducing your hard won gains.
I have actually done what you are wanting to do, but I can't promise that it will work for you or anyone else for that matter. It is possible that I was only successful because I had 160lbs of excess fat to use for fuel.
You didn't post your height, weight, BF%, LBM to give us an idea of what you may need to do. These would be helpful.
The way I did it was to go on a very strictly controlled carbohydrate plan - also known as a ketogenic diet. This stabilized my blood sugar, allowed my body to quit producing so much insulin, and allowed me to burn fat. The extra LBM that I built was just a nice side benefit.
I started by cutting my
carbohydrate intake to about 20g a day. I didn't count calories, I ate all the protein and fat that I wanted. This meant that I was eating whole eggs, sausage, steaks, hamburgers without buns. I ate two big salads each day. After a few weeks, I started substituting other green veggies for the salads - the high fiber, low glycemic ones. Examples are spinach, asparagus, broccoli, green beans. I ate until satisfied, not "full" or "stuffed", just until no longer hungry. I ate every 4 - 6 hours. If I found myself getting hungry more often or the fat loss stalling, I increased my fat intake. The nice thing about the ketogenic diet is that it seems to be muscle sparing - unlike simple
calorie restriction.
After a 4 month bulking cycle, my abs were almost gone again so I went back on it. This time, not quite as strict. I kept supplimenting with protein shakes and kept using dextrose post workout. This put my carbs at around 100g each day. It works for me, I'm still burning fat, just more slowly.
Since from your description, you aren't really overweight. Not being able to see abs doesn't make one obese you know. Look back at the old movies and fight clips from the '70s - see Chuck Norris without a shirt - definitely fit, not a fat man - but no abs - even some jiggles.
If you really want to do it, I would suggest that first you get weighed, keep track of BF% and LBM. Then keep an honest accounting of exactly what you eat and when. After you have done this, start gradually decreasing your carbohydrate intake and increasing fat intake very slightly (preferably healthy natural fats, not trans fats or hydrogenated oils). Keep lifting, do uour cardio. Cardio is very important for cholesterol control, BF control, and is just plain old good for your heart. When you reach the point that BF is going down and LBM is going up - that is where you need to stay.
Some people, me for example - are carb addicts. We have terrible carb cravings. I had to do the strict Atkins style induction (20g complex carbs each day) for a couple of weeks to get past the cravings.
That is just how I would do it because it worked for me. Others may have a better idea that will work for you.
Since you are a hard gainer, you may just have to increase your cardio AND feed your workouts (including your cardio) - eat some protein before workouts, eat some carbohydrate and protein right after them. Perhaps if you simply eat most of your carbs closer to the time of your workouts it will work better for you. Keep the simple carbs immediately post workout with some protein to feed those muscles - and keep it away from going into body fat storage.
Hope you find the way that is going to get you to your goal.