It is possible to create a TE clay generator. It is also a very slow and finicky process.
For the smallest TE clay generator, I need two pulverizers (one for sand, one for ingots), an induction furnace, and a cyclic assembler. At three seconds per Pulverizer operation, 25% chance of slag per smelt, and 1 slag per clay ball, I average one block of clay every 96 seconds. Check my napkin math on that one?
1 block = 4 clay = 4 slag = ~16 Induction Smelter operations = ~32 Pulverizer operations
On a chunk loaded server, that's an average of around 14 stacks every 24 hours, and around 4.5 days to fill a barrel. I'll need to generate dirt, and a way to limit how much dirt is supplied to the Cyclic Assembler at one time if I want to leave it for long periods. The 17 other inventory spaces of a Cyclic Assembler can hold enough dirt for around 6 hours of operation of this smallest generator, making it difficult to keep manually refilled. A hopper or item duct set-up will eventually fill all spaces with dirt. I'll need a more selective mod or a partner.
Taking that to a scale that could fill that barrel in (on average) 6 hours 50 minutes (similar to an igneous extruder producing obsidian) would require a hefty resource and power investment. I'll need those 16 Induction Smelter operations to happen every six seconds. An Induction Smelter on Dust takes one second, so that's 3 Inductions Smelters. I need to provide 32 Pulverized Iron per six seconds, meaning 16 Pulverizers working on metal. I'll need to supply 16 sand every six seconds, so 8 pulverizers making sand. To supply 16 Cobblestone every six seconds would require 6 Igneous Extruders. I will need two cyclic assemblers and an aqueous accumulator to perform the crafts. Same challenges apply to the dirt supply. I'm using 1080 rf/tick, and have a lot of materials invested in machines.
So, in conclusion, I really suggest looking elsewhere if you want to use hardened clay as a major building block in vast quantities. Magical Crops with Thaumcraft Golem automation seems like the best bet. It would be nice to see a machine that could come close.