Deep in the money Covered Calls with high premiums

Let's start with a small correction to your second example. The underlying is 354.20 and the 315 call is 51.05 . That means that the intrinsic value is 39.20 and the time premium is 11.85 . Still quite hefty but not realistic. As mentioned in the comments, these are closing quotes and will not be reflective of the actual market since illiquid options may have traded minutes, hours or even days before the closing price of the underlying. See real time prices.

Based on the implied volatility of the 355 call, the premium for the 315 call should be closer to 10 points lower. There is no price inefficiency here.

Choice of strike price is a trade off between risk and reward:

• Selling a lower strike provides more premium, more downside protection with a lower potential profit.

• Selling a higher strike provides less premium, less downside protection with a higher potential profit.

The optimal result is attained by knowing what share price is going to be at expiration. Since that can’t be known, you have to find a balance between the two choices. What's your motivation for the position, fear or greed?

Covered calls are synthetically equivalent to short puts. If you have the approval and the numbers are good, selling the put has the potential for fewer commissions and incurring fewer B/A spread costs.

A covered call has an asymmetric risk profile with a limited upside while bearing most of the downside risk. So unless you have some secret sauce that gives you superior timing and selection, I'd suggest that you consider strategies with a better risk profile (vertical or diagonal spreads). These would have a modestly lower profit potential while greatly reducing the risk, shifting the R/R from asymmetric to much closer to balanced.

An alternative approach to these strategies would be to use high delta ITM calls as the long leg, reducing the risk should a collapse in the underlying occur (Google "Stock Replacement Strategy" and "Poor Man's Covered Call").