What accessory do I need to connect two external monitors to my early 2011 15 inch MacBook Pro?

The built-in display continues to work, and you may get very good results using the built-in as an "extra" display. There is no need to close the Mac to use an external display.

everymac.com says:

supports AN external display at 2560x1600 and passes an audio signal

For the second external display, you would need a USB display adapter. USB display adapters are s-l-o-w to update, and using a mouse or trackpad with it can be torturous.

http://eshop.macsales.com/item/NewerTech/VIDU3DVIA/

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Buy the item I linked (the blue underlined text in my response is a link to the product offered by a popular, Mac-centric vendor).

Plug it into a USB port.

Install the Mac Driver.

connect your second external display to it.

You will then have the best solution possible for the Mac that you own.

If that is not a "nice enough" answer, you will need to convert your current MacBook to CASH (by selling it) and buy one that supports more displays "out-of-the-box".

USB displays are slow to update. If you do not need to use a mouse or trackpad with it, that may be tolerable.

Using the ThunderBolt port for additional displays is an intriguing idea.

But on your particular Mac, there is only ONE ThunderBolt port, and daisy-chaining of external displays is not supported on any Mac at this writing. If you wanted to buy an external stunt-box with an additional display port on it, that is possible, but none have been offered at reasonable prices.

A newer Mac with more display support is your best option for really fast-updating displays. All others are compromises.