The best time to prune Burning bush (Euonymus alatus) is in early spring before it begins to leaf out. This allows for you to see the shape, helps with the healing of pruning cuts, and can minimize pests. There are two ways to prune an older, larger shrub.

First: General pruning can be done at any time. This is the removal of dead, dying, broken, or injured stems/branches. Also look for any stems or branch stubs that are intertwined or rubbing against each other. These should be removed at the base of the shrub.

Renewal pruning (or thinning) means taking out only 1/3 of the oldest branches. The next year prune out 1/3 of the oldest, and then again the following year. This form of pruning can be done every year. The largest stems will be the oldest, so remove them at the base at ground level. This leaves the shorter, younger, more vigorous stems. Renewal pruning allows younger, fuller branching to replace the taller, older, thicker stems over time resulting in a healthier shrub.

Rejuvenation pruning is the most severe form of pruning. This means cutting the entire shrub down to ground level using very sharp shears or pruners. Leave at least 3 inches above ground. New growth will form shortly thereafter. This heavy pruning leaves you with a smaller, “younger” plant. Rejuvenation pruning should NOT be considered if the plant is stressed, diseased, plagued by insects, or in very poor health. Sometimes this pruning method seems too harsh for homeowners, so they opt for renewal pruning.

Here is CSU's information on pruning shrubs. You do not have to be concerned about 'flowering' on the Burning bush, but the pruning information is the same.

https://static.colostate.edu/client-files/csfs/pdfs/PruningFloweringShrubs_619.pdf