According to :
Time to check all of the connections. Especially the neutrals and grounds.
Lights dimming when a motor starts is classic loose neutral.
If your not comfortable working inside the electrical panel hire some one.

Lights _brightening_ is a sign of loose neutral.

Lights dimming is more of a sign of a loose hot.

When you say all the connections, do you mean all the connections in
the panel or all the connections that are on the boiler circuit (which
is a separate circuit)?

I can easily pop open the panel after shutting down the breaker outside
the house (the socket has a breaker) and make sure all the grounds and
neutrals are tight. If they are tight, then what should I check next?

By the way, the issue is dimming and not flickering. I suspect the
latter would be worse although I am not sure.

Likely a poor connection in one of the main hots. Loose connections
don't always show as flickering (flickering is arcing).
--
Chris Lewis, Una confibula non set est
It's not just anyone who gets a Starship Cruiser class named after them.