You can’t avoid self employment tax. 

You would need to set up a multi member LLC and then have your SSN on your 1099-Form replaced with the EIN for the business. The business will still have to pay taxes as an S corporation depending upon your state.  

A single member LLC is treated as sole proprietorship on the tax return nd the net profit from that is self employment income subject to self employment tax. 

One way to reduce self employment tax is to elect to treat your LLC as an S Corporation on your tax return.   You can classify some of your income as salary and some as a distribution. You'll still be liable for self-employment taxes on the salary portion of your income, but you'll just pay ordinary income tax on the distribution portion. Depending on how you divide your income, you could save a substantial amount of self-employment taxes just by electing to treat your LLC as an S Corporation on the tax return. 

How an S-Corp Can Reduce Your Self-Employment Taxes

The SE tax can't be avoided. As a single member LLC the business is considered a disregarded entity for tax purposes.  The SE tax is the employer paid side of your social security and Medicare tax. So as a disregarded entity you pay it.

As an S-Corp and with you as an employee of that S-Corp, the corporation is the employer and the corporation pays that tax. So it gets paid one way or the other.