"If I were to ask the Chief to your left if you could tell the truth would he say '

A'?"
EXPLINATION:
I'll notate them as:
Truthie: T
Falsie: F
Mixed: M
So there are 6 possibilities:
seat:| 1 | 2 | 3 |
------------------
a: | T | F | M |
b: | T | M | F |
c: | F | T | M |
d: | F | M | T |
e: | M | T | F |
f: | M | F | T |
I'll translate "Chief to your left" to a number for ease.
Question 1: to Chief #1: "If I asked #2 if you could tell the truth would he say '

A'?"
Now here is the third crux: the Truthful Chief ALWAYS tells the truth and the False Chief ALWAYS lies so if either of them don't know then they can not answer.
So if #2 is the mixed chief then you will get silence to Question #1. Giving you three possible answers to the yes/no Question which is how you can get 6 possibilities out of only 2 Chief's answers.
Q1 answer #a: Silence:
seat:| 1 | 2 | 3 |
------------------
b: | T | M | F |
d: | F | M | T |
Question 2#a :
If silence to Q1 then you ask the same question to seat#3, 'to your left now means seat #1 so: "Seat #3: If I were to ask #1 if you could tell the truth would he say '

A'?"
If DA = yes
seat:| 1 | 2 | 3 |
------------------
b: | T | M | F | = DA
d: | F | M | T | = JA
If DA = no, same result
seat:| 1 | 2 | 3 |
------------------
b: | T | M | F | = DA
d: | F | M | T | = JA
I'll explain:
---If DA = yes-----------------
seat:| 1 | 2 | 3 |
------------------
b: | T | M | F | = DA: 3 can't tell the truth, 1 would be honest and say no translating to 'JA', 3 then lies about weather 1 would say '

A' and says yes translating to DA.
seat:| 1 | 2 | 3 |
------------------
d: | F | M | T | = JA: 3 can tell the truth, 1 would lie about that and say no which translates to 'JA', 3 is then honest about weather 1 would say '

A' and says no translating to JA.
---If DA = no-----------------
seat:| 1 | 2 | 3 |
------------------
b: | T | M | F | = DA: 3 can't tell the truth, 1 would be honest and say no translating to '

A', 3 then lies about weather 1 would say '

A' and says no translating to DA.
seat:| 1 | 2 | 3 |
------------------
d: | F | M | T | = JA: 3 can tell the truth, 1 would lie about that and say no which translates to '

A', 3 is then honest about weather 1 would say '

A' and says yes translating to JA.
Great so what if Q1 doesn't come back in silence:
Then you know #2 is NOT mixed
Let's remember if the answer comes back DA it's a lie, JA it's honest
if Q1 answer = DA (lie) then we know #2 is NOT Mixed and #1 is not Truthie so there are 3 possible scenarios:
seat:| 1 | 2 | 3 |
------------------
c: | F | T | M |
e: | M | T | F |
f: | M | F | T |
Then you ask #2 "If I were to ask #3 if you could tell the truth would he say '

A'?"
If silence then #3 is Mixed and:
seat:| 1 | 2 | 3 |
------------------
c: | F | T | M |
If DA then #2 is Falsie and #3 is NOT Mixed:
seat:| 1 | 2 | 3 |
------------------
f: | M | F | T |
If JA then #2 is Truthie and #3 is NOT Mixed:
seat:| 1 | 2 | 3 |
------------------
e: | M | T | F |
if Q1 answer = JA (truth) then we know #2 is NOT Mixed and #1 is not Falsie so there are 3 possible scenarios:
seat:| 1 | 2 | 3 |
------------------
a: | T | F | M |
e: | M | T | F |
f: | M | F | T |
Then you ask #2 "If I were to ask #3 if you could tell the truth would he say '

A'?"
If silence then #3 is Mixed and:
seat:| 1 | 2 | 3 |
------------------
a: | T | F | M |
If DA then #2 is Falsie and #3 is NOT Mixed:
seat:| 1 | 2 | 3 |
------------------
f: | M | F | T |
If JA then #2 is Truthie and #3 is NOT Mixed:
seat:| 1 | 2 | 3 |
------------------
e: | M | T | F |