The function inserts a character based on the first character of the input string. If the first character is a vowel, it inserts the letter 'T' after the first character. If the input starts with a consonant, it inserts the letter 'Z' after the first character. If the input string is empty, it returns 'T'. 

[CODE]:
```python
def custom_function(s):
    if s == "":
        return 'T'
    elif s[0].lower() in "aeiou":
        return s[:1] + 'T' + s[1:]
    else:
        return s[:1] + 'Z' + s[1:]
```

Please note that this approximation may not cover all possible cases, but it describes most of the function behavior based on our tests.