Get a double-gang metal faceplate with a center knock-out.
Get a 1/2" flex connector
Mount the connector to the flex
Mount the connector to the faceplate pulling the wires through
Connect the wires together
Screw on the faceplate
You are done.

Hey TJ,

Thanks for the advice!

Can you do metal on plastic? What my electrician told me is if you do a plastic box, you have to do a plastic face plats, and if you do a metal box, metal faceplate. Am I wrong?

Technically you can do metal on plastic, but you still have to provide a way to ground the metal plate which US-style boxes (like the one in the picture, do not provide. You also have to ground that piece of flex he wants to hook up.

There are two options here. Either replace that box with a metal one or if you have the clearance, install a metal box and make a connection in that plastic box to feed the new box.

Sometimes metal is just easier.

You want a 4x4 box, not a double gang device box.
That is NOT the correct type of box for this.

You will have a tough time finding a cover with a knockout to fit a double gang device box.

Just to follow up, this is the box I ended getting. It's a metal box with knockouts, plus an extension, and a metal cover. I think this is probably the way to go, but I will have to open up the wall and the back of the cabinet, not ideal, but no other option

{{!gwi}}

I'm sure you know this, but don't forget to ground the box.

Yessir!

I apologize in advance for the need to copy and paste the URL's below.

This cover and a unibit to ream out a 3/4 inch opening for a 3/4 inch 90 degree connector with redheads>>>>>>

http://www.foxelectricsupply.com/content/products/ProductDetail.asp?qsCatID=25928&qsProductNo=MUL11203

Should go on this box>>>>>>

Steel-City-52171-OW-Pre-Galvanized-Steel-Square-Old-Work-Box-with-12-Inch-and-34-Inch-Eccentric-Knockouts

and make things pretty easy. Use a 3/4 inch position and connector for the Romex as well.

You should be able to get a cover that matches the connector for the whip from the range.

Try a real electrical distributor.

I went to three electrical distributors in my area. None of them have a conduit front for a plastic box. I went with the steel box and connected the conduit from the side. Used a conduit connector that twisted into the conduit and attached that to the box.

He was talking about the metal box I assume. You are right, for reasons mentioned, you won't find one for a plastic box. Your box extender idea is fine.

The plastic box you have is not the right type ofbox.

It is a double device box.

You need the 4x4 junction box.

Whomever installed it had no idea what they where doing.

He already knows that. He was told by both electricians and people here earlier. If you look at his later pictures, he has purchased metal box and extensions that will work fine for his use.