Experiments with the boulders of ice on the ground establish that it's more prone to break the more heat is applied. One of the catfolk is a jeweler, who compares it with how glass explodes if you put it in flame unevenly or touch it with a cold tool. Further experiments are focused on ways to apply small amounts of heat deep inside the ice without going into a dangerous place oneself.
Touch the ice with a hot stone rod? It works but you can't get very far with it. Make a hole and lower in a stone holding catfire on a chain? Sure, and you can scoop the water out with buckets. The jeweler points out that the water is analogous to hot glass and maybe shouldn't touch the ice... and experiments bear out that concern. So they'll make a tiny aqueduct to get the water off the mountain?
A team tries putting water into a piston and spraying it into the air, where it freezes and blows away. That seems easier, as long as the snow isn't going to be a problem somehow - a crew is assigned to make snow using that method, upwind of an unused part of the boulder pile, continually for a cycle.
Would it be possible to melt the ice by spraying molten ice on it? Yes, it's certainly possible, with the water getting pushed back out the hole to be heated again, but some of the water is disappearing... and then the boulder cracks and water gushes out. Ultimately, this way is slower, more complicated, and more unpredictable than inserting a flame-covered stone.
Back to the simple method, now with a shaft that spins a pump to push the meltwater out to the top.
Problem: if a catflame is incidentally covered in water, it boils a bubble of steam that can crack the ice. Solution: put the catfire into a hollow inside the rod, where it can't get wet. Problem: now the rod explodes. Solution to that: leave a tiny hole for hot air to get out while sealing the flame.
Problem: if water gets at the pump shaft, it can freeze and then it's hard to unstick the shaft without breaking it. Solution: use a bubble pump, like a tiny version of the really big pumps in Kef, using a tiny turbine to push the air, like in Audron's design.
The next challenge is connecting segments of rod together without leaking air or water, but actually making the tube has been good practice for that.
They confirm that boring an upwards sloping drain from the bottom works too. Getting them to meet is hard but it's fine if they need multiple attempts; they're going to need to remove a whole lot of ice anyway. They confirm that they can line the L shape with stone and gradually expand it.
Time go to on top of the mountain and start! With a ship like a bell. Everyone will stay inside the whole time. Normally the ship will be frozen (and if the ground collapses it's obviously fine). They go up by freezing the separate 'floor' and then everyone hauls the ship upward by rope handles. Going sideways is similar.