Question
I live in Brooklyn, NY. When is the best time to prune my English Holly and my Golden Euonymous bushes

Answer
John,

Pruning English Holly
Step1
Prune in December. It is good for the tree, and you can use your cuttings to decorate for the holidays around your house.
Step2
Avoid shaping your holly too much. You shouldn't expect holly to be a boxed hedging or topiary. Prune the bush in a way that continues the natural growing shape of the plant.
Step3
Start from the inside and work your way out. Cut away branches that are rubbing on each other or any branch that looks diseased or dead. Trim branches just above a new leaf bud or all the way back to a main branch. You can cut a great deal off of your holly to thin it out. Doing this will make your bush full and lovely again in a few years.
Step4
Avoid using hedge trimmers. Taking the ends off branches and leaves doesn't help your holly grow and creates an unappealing boxy look.
Step5
Remove all the trimmings. Salvage some for decorations around the house but dispose of others. If any are diseased, keep them away from your holly bush.

Euonymous bushes:
These should be pruned in early spring.  There are many varieties of Euonymous bushes.  If yours is a "burning bush" type then definitely the early spring.  The short green shrubby ones can be trimmed almost anytime.

Burning bushes or Euonymus alatus normally only have their growth on the end of the branches, this is why you see little to no growth on the inside of the plants. 6-8 feet is the maximum that these shrubs will grow, but you can prune them back to whatever height you wish to keep them at.
Spring is the best time to do a rejuvination pruning. In the spring, you can cut it back as far as you want. You should be aware that you may not see any growth next year, but the following year, the plants will releaf as normal.

Sijka