Words aren't broad or slender. Letters are.
For a 1st conjugation verb like téigh, the Tips & Notes say that you have ann as the "broad ending", and eann as the "slender ending" in the present tense, but that really only applies to a stem that ends in a consonant, like ól or dún or ith or bris - you have to use the correct ending based on that final consonant - you can't put ann after ith or bris because of the slender i before the final consonant, and you can't put eann after ól or dún because of the broad ó and ú before the final consonant. If the final consonant is broad (because there is a broad vowel before it), you use the "broad ending", if the final consonant is slender (because there is a slender vowel before it), you use the "slender ending".
But in the case of téigh, the stem is té, and you don't have a final consonant that needs to be kept either broad or slender, and there is no rule preventing a slender vowel é coming next to a broad a, so you get téann rather than téeann.