Good morning electrical geniuses, here's a problem for you.

I have a piece of equipment that requires a 220v connection, and I do not have the ability to run a 220v line from the breaker. Is there a Y adaptor I could buy to combine 2 different phases of 110v to create a 220v power strip? all of the 110v outlets that I have access to are 20a breakers, so would this new 240v line give me 40a (assuming the hardware and wire gauge can handle it)?

The hardware is going to draw something in the neighborhood of 3200w/15a @ 220v (3200w/30a @ 110v), so if I combined 2x 110v/20a with a high gauge Y adaptor, I shouldn't have any problem as long as the wiring was correct?

I'm not an electrical engineer, I'm a mechanical one... Help, please :)

Edit 1: @ThreePhaseEel
Here are photos of the electrical panel. The room with my specific panel is locked, but this is one of 3 identical rooms per floor, 4 floors. The room next to mine is occupied by someone who only has things plugged into the walls in the room, so if our bathroom receptacles are connected It won't matter.
[![Breaker Panel for the Posted Description](https://i.stack.imgur.com/ddgV8.jpg)](https://i.stack.imgur.com/ddgV8.jpg)
[![Breaker Panel Description](https://i.stack.imgur.com/yc9aq.jpg)](https://i.stack.imgur.com/yc9aq.jpg)
[![Overview of Whole Panel](https://i.stack.imgur.com/CfWjA.jpg)](https://i.stack.imgur.com/CfWjA.jpg)