A 6-7 minute climb? Something like a mile or so (2km)?

Try standing up and pushing in a HIGH GEAR the whole climb. I don't mean 1 or 2 cogs bigger, try the big chainring and something in the middle cog. PUSH! and push hard. Repeat 2-3x per week.

Once your heart and lungs catch up to the demand you're putting on them you'll get faster. Think of it as being a stair climbing machine.