The sandwich concept is great, but you're forgetting something big:
You might have offended or insulted or hurt the other person, and they may have taken something you said personally. If that's the case, you have to take care when you respond.

I have a friend I talk to nearly every day who sometimes gets testy with me. Everyone does that; we are human.

The first thing I do is stall for time. I need to make sure I'm not going to take it personally and react in a defensive way. I don't waste time saying, "What's your problem?" or "Whoa, calm down!" or anything like that. I will sometimes say exactly what's on my mind - "It seems like the conversation is taking a turn" or something similar.

The next thing I do is try to figure out what I did. Because 100% of the time I have contributed in some way. I might not have done anything on purpose - I might have hit a sore spot, or I might have miscommunicated something. Why it happened doesn't matter. It happened. I jump right in and correct it, clarify what I was saying if I was unclear, or say, "should we talk about something else now" if the topic is triggering.

And never make excuses. There is a big difference between saying, "I might not have been clear" and "you should know what I'm talking about." And that includes blaming a language difference - hello, Google translate! Dismissing or diminishing the hurt you have done is just adding insult to injury. Watch humor as a way to deflect, also - you might think you're witty, but you might actually be cruelly making fun of someone else's pain.

Think about it this way. If you step on someone's foot, even if it's an accident, you're going to apologize. If you make an excuse and say, "I didn't see you there," dismissing the injury you did, think about what you're telling that person. I didn't see you there = you're not worth me noticing = you don't matter. That's horrible!

If you feel criticized, especially when you're not actually being criticized, like if someone corrects your error or expresses a different opinion than the one you have, think hard about what you might have done to cause it. I'm not saying go into an avoidant free fall, where you blame yourself for living. Don't personalize it. Pull it apart and see how you contributed. Fix that, and you will show that you care for the other person.

If you dismiss it, or whine about how you feel so bad that you made the other person feel bad, so even though you did the original injury now you're requiring them to take care of your pain, that's callous, which is the opposite of compassion. It's very selfish. You're saying that your own pain is what matters, not the other person.

Pay attention if it's the tone that makes you feel like someone has "criticized" you. If it feels critical when it's actually not, when they're just pointing out an error or giving their opinion, you've probably stepped on someone's toes, whether your realized it or not.