Thanks for all the input!
Luckily the dripping element was outside the area over the pizza
(mushrooms, onions, roasted garlic, pesto) -- and it was delicious --
with no short term health effects except extra pounds. The problem
definitely wasn't related to food splatter -- the element looked like
someone had gone at it with a cutting torch. While the fire was going,
the display showed PF (power failure). It was only after I threw the
breaker that the element stopped burning; once the breaker was reset,
the dislay panel worked as though nothing had happened.

dpb was correct in the sequence of events:
1) the tech (under a service contract with Century Service Systems in
Florida) arrived one week later
2) examined the element
3) came back 2 weeks later with a new one which he installed and
determined was not working (by touch)
4) removed the rear cover (exposing the control board)
5) assumed it was a bad control board without making any test
6) left

One week later I get a call from the office telling me that a new
board will be $250 (not covered under the service contract).

I do not use the broiler very often so I'm leaning towards leaving it
as is, HOWEVER, the 'other half' wants it fixed (even if she's not
willing to pony up the $250).

I do all the cooking and can't stand the flat top stoves (except for
clean-up). No choice but to use electric (condo). I especially hate
having to reach over the tops of the pots to turn the 'burners' on/
off.
As for the features (self clean, timer, digital display) -- all things
I can live without -- but I'll admit that in ant/bug ridden Florida,
not having to deal with cleaning out the area under the stove-top is
a definite plus.

Based on the info, I think I should get the service company to send
out a tech who can determine whether there is current to the element
and if there is an internal fuse. I'm also checking out other sources
for control boards.

Thanks again for everyone's responses.