This is a great idea. I grow Pineapple Sage and a variety of Basil plants each year that get 3-6 feet tall, and they have interesting, woody trunks, and respond well to pruning. I've seriously thought about doing this.
In the 2017-18 winter, I cut a big (ground grown) Pineapple Sage down to ground level after a hard freeze seriously scorched it, and buried it in mulch, and when it came back in the spring, it really took off. This particular Pineapple Sage grows into a big 5' ball, and in the last few weeks has started putting out red flowers. I'm going to try the same thing with some of the big basil plants I currently have in pots. Pineapple Sage seems to really like our climate, while other sage varieties I've grown have been much more tentative.
Currently growing Sweet Basil, Cinnamon Basil, and Thai Basil in 15 gallon pots, and each of those seem to really like the climate here. The basils have been blooming for about two months. I cut all the blooms off half the plants (to keep them from going to seed), and when they start to bloom again, I cut the blooms off the other half of the plants, to continue have flowers for the bees, and to keep them from dying after going to seed. I don't know if the basils will come back, or not. Had numerous volunteer basils jump up in pots and in the garden from plants that went to seed in pots in the 2017 season. I'd love to grow rosemary, but I manage to kill them, every time.