Would Frozen Fortress kill a 1 might unit that controls it at the start of the turn?

Yes, Frozen Fortress will kill a 1-Might unit at that battlefield.

According to the ability text, "At the start of each player's Beginning Phase, deal 1 to each unit here." When that 1 damage is dealt to a unit with 1 Might, the unit has damage marked on it equal to its Might, which results in it being killed during the subsequent cleanup check (RiftJudge FAQ #10214).

The fact that the unit controls the battlefield does not provide immunity to the damage dealt by that battlefield's ability. As stated in Rule 187.6, the battlefield's controller (or the turn player if it is uncontrolled) is responsible for the ability, and the damage is dealt to all units at that location regardless of who they belong to or whether they control the battlefield.

Sequence:

1. Beginning Phase Begins: The "at the start of each player's Beginning Phase" trigger activates.
2. Chain: The ability is placed on the chain.
3. Resolution: The ability resolves, dealing 1 damage to the 1-Might unit.
4. Cleanup: The game checks for lethal damage. Since the unit has 1 damage marked and 1 Might, it is killed.