What you want is a variable frequency drive, or VFD. This is a commodity part, easily available from a wide range of manufacturers and distributors. At the power levels you're describing, many are specifically designed to take single-phase AC in and put out three-phase AC out to a motor.
A quick Google search found these:
Now, if you actually want to design such a thing, that's much more complicated! You'll need to know how to design a rectifier, both electrically and thermally. You'll need to know the same about an inverter. You'll need to size the capacitors properly. You'll need a precharge. And that's before you even get into how you control a motor by feeding it voltage and frequency. This is very much not a trivial endeavor; there are several billion-dollar companies who do this for a living!
Not to say you can't or shouldn't do it. Just know that it's an investment of months or years into making something that you can buy for $300. If you succeed, you'll be much smarter! Just make sure you understand that that's your goal when you start, and not necessarily to be cost-effective.