Crazy hungry. Do I eat more or not?

I'm a week into the Strength program, after using FitnessBlender on and off for years on Youtube and the website. I've always loved strength training and I'm really enjoying just clicking open the calendar and doing the next video, without having to think about what to do next.

What I've noticed now, a week in, is how much my appetite has increased. I'm vegetarian and generally eat very clean. No fried foods, mostly organic. I drink loads of water, a good 2 litres a day. I have oatmeal with fruit and almonds at breakfast, rice and lentils at lunch, that sort of thing. I did add a protein shake to my morning routine which has helped my recovery time and reduced the jelly feeling in my muscles after a workout. But by 11:30am I'm super hungry for lunch, and my 3pm I'm ready for dinner. If I eat an early dinner, I'm hungry at bedtime. My usual meal routine clearly isn't enough now.

BUT! I also have perhaps 15 - 20 lbs of fat to lose. (I don't own a scale, so I don't know for sure...I'm taking measurements and photos for before and afters....already my clothes are feeling looser) While I want to eat to support muscle development, I also want to lean out. But I also don't want to be shaky and hungry to the point of distraction, or not have the energy for my workouts.

Suggestions please!