You don't want to just start ordinary paint over the varnish. You'll have
to prime it first.

From the Zinsser web site
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Topic - B-I-N for Kitchen Cabinets(1) Problem:

In redecorating my kitchen, I plan to paint the cabinets which now have a
clear high-gloss wood finish. Should I prime with B-I-N?

Solution:
B-I-N is the best primer for the job. It has great adhesion to glossy
surfaces and dries to a hard abrasion-resistant base coat in 45 minutes.
(Other primers can require up to 14 days to cure completely .) Before
priming, be sure to remove oily greasy deposits by wiping down the cabinet
surfaces with mineral spirits, followed by washing with a 50:50 solution of
ammonia and water.

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they claim it will go on w/no sanding but it can't hurt to sand them lightly
w/150-220 grit paper first.
After the primer is cured you can't paint on it w/whatever you like.

ml