How to improve pedalling strength/hill climbing

Recently, I started running in addition to cycling. I like working my muscles differently than cycling does and I can get my heart rate up more easily I find. (kind of a guess; no HR monitor)

I warm-up on uneven flat ground (ie, flat but with some inclines).

Then, I sprint up a good-size hill. I use all my muscles (arms, core, all leg muscles) to vault myself up the hill, maintaining the high speed to the top. I walk at a quick pace for 10 seconds, jog down, walk quickly 10 seconds, repeat.

Once I can't stay fast all the way to the top, I do a few more then stop.

I eat a lot of protein after, sleep a lot, and a week after almost no riding and only a couple of these sessions, I attacked two hills on my ride to work that used to exhaust me. I hit these hills consecutively (I used to wait an extra stop-light as a break because I get no warm-up or downhill for attack speed) and got to work with no adrenaline rush. I felt great and didn't crash at all. In fact, I got home and did one of these work-outs, and when I did a 90 km high speed ride (the Donut Ride) on Saturday I was going faster than I had on previous long-distance rides.

Hope it helps. Any advice is welcome as I'm NOT a trainer. This is from my personal experience.

I'm also vegetarian, and this surprises people who think veggies are weaklings with no muscle ;)