OK, for growing 100 lettuce plants I would probably go with a NFT system with a minimum 100 gallon
nutrient reservoir, and even larger (150 gallon) nutrient reservoir if they are large lettuce plants. You could use a water culture system, but with that many plants, it would have to be very large, and probably need to hold at least 1,000 gallons or more to be big enough.
As for the mint well you didn't really answer my question about what variety and how big they are expected to get. So I have to assume they will get as big as the ones I'm used to that we had growing at the condo complex I used to live at in Calif. Those mint plants grew to be hedges. 5-6 feet tall, and just as wide. And that is with the gardeners trimming them back every week. I have no Idea what your planning to do with this many huge mint plants. Even if you want to sell it, the stuff grows so fast like a weed, and gets huge. You could probably expect to harvest at least 5-6 ponds of mint from each plant weekly. 100 plants should yield at least 500-600 pounds of product weekly.
There is no doubt that I would use a
drip system to grow large plants like this. Each plant having it's own 5-10 gallon bucket for the roots for plants expected to get this size. The size of these plants will require a much larger nutrient reservoir. Minimum 5 gallons of water volume per plant (100 plants) totaling a minimum 500 gallon nutrient reservoir but more realistically closer to 750-1,000 gallons total for the 100 plants. However I would break it up. Instead of one 750-1,000 gallon nutrient reservoir. I would prefer to use four 250 gallon nutrient reservoirs, that feed 25 plants each. I seriously don't think you would have room for this large of a setup on that roof. 100 large mint plants like this would probably take up 1/2 acres of space. The mint plants we grew in Calif never died, they lived for decades. They just got trimmed back every week.
As for your statement that you want to preserve/harvest the roots as well. Again I'm at a loss because you didn't answer my question about when you planing to harvest/pull the mint plants. As I mentioned, they won't die, they'll live decades. But if harvesting the roots is a goal, I would never let the plants get so big to begin with. Maybe even use a different variety that doesn't get so big, and/or harvest before they get very big. This way instead of using a drip system, I could construct a low pressure
aeroponic system to grow them in. That way I can keep the roots clean and not need to separate them from the growing media when I harvested them. That not only saves labor separating it, but money as well because you don't need to buy all that growing media. And because the roots aren't in a growing media, their clean and much better product.
But again everything depends on the questions I asked about the variety and how big they are expected to get. Without that info it's just working backwards and spinning wheels.