Does the Fed "monetize the debt" by selling bonds?

I can understand how the Fed could increase the money supply by
manipulating the federal funds rate or the reserve requirement.

But recently I've heard it suggested that the Fed can also "monetize
the debt" (i.e., increase the money supply to offset higher interest
rates caused by government borrowing) by selling bonds.

This is what I don't get:  it seems that this particular "cure" is the
exact same thing that caused the "disease":  selling bonds.

Here's my thinking:

1.  The government plans to spend more than it will collect in taxes. 
Therefore, it sells bonds to the public.

2.  This "crowds out" other borrowers who must now pay the higher
interest rate arising from increased demand, or not borrow.

3.  To offset this higher interest rate, the Fed buys bonds from the
public, increasing the supply of money in the economy.

So you have the government both buying and selling bonds?  It seems
that would be working at cross-purposes.

What am I not getting?

First, make sure you're clear that the Fed can only sell US Treasury
Bonds that it had previously purchased... it's selling "old" bonds,
not "new" ones. Only the US Treasury can issue new debt.

Also, I think you made a semantic mistake in your question. You state
that you've heard "the Fed can also monetize the debt by selling
bonds." Later on, you say "Fed buys bonds from the public, increasing
the supply of money in the economy." I'm assuming you meant both
sentances to read that the Fed is raising the money supply by buying
bonds (selling bonds would tighten the money supply... it would also
reinforce the effects of the US Treasury's bond sales).

It sounds to me that your question is thus: Can the Fed take action to
lessen the negative impacts of US Treasury (over-)borrowing by
essentially buying the newly-issued US Treasury debt?

Short answer, yes. This is fairly common. If the US Treasury takes
action that results in monetary tightening, the Fed will do what's
necessary to bring the money supply back in line with the Fed's policy
goals. This may take the form of the US Treasury issuing/selling new
bonds while the Fed is buying old Treasury bonds off the market.

However, the whole reason we have an independent Fed is to prevent
wholesale "running of the printing presses", where newly-created money
is used to subsize government profligacy. The printing press problem
was common in third-world nations; governments print new money to pay
for social spending. When this gets out of hand, rampant inflation
takes hold.

The Fed's role is to prevent inflation, so it will not release too
much money into the money supply in response to government behavior.
Also, the Fed does have statutory limits on how much it can grow the
money supply.

As an aside, there's a term for the opposite of the process you
described. If the monetary authority is releasing too much new money
(example: China, to maintain the currency peg), the government can
step in to MOP UP the excess liquidity (China sells long-term
government bonds). This is referred to as mopping up liquidity.