Adjectives most often come before the noun they describe, but they can be placed after the noun as explained in this article [Extracts]:
 Some adjectives ending in -able/-ible can also be used after nouns [example:]
It is the only solution possible.
Book all the tickets available [...]
 Adjectives come after words like something, everything, anything, nothing, somebody, anywhere etc.
I would like to go somewhere quiet.
I heard something interesting today [...]
 In most expressions of measurement adjectives come after the measurement noun.
ten years older
six feet deep
two miles long [...]
 Verb + object + adjective
Adjectives can be placed after the object.
You make me happy.
Can you get the children ready for school?
In addition, the adjective may be placed after the noun in poetic or literary constructions not usually seen in ordinary English, as in
Sedately sits the miller stout,
Watching the village roundabout
[Piece of a poem: here "the miller stout" means the stout (fat) miller and "stout" is the adjective coming after the noun so that "miller stout" can rhyme with "roundabout": this is an example of poetic license]
This can also occur in some idioms:
For the US Team at this Olympics, the gold medal proved a bridge too far (or a hurdle too high)
[Here "a bridge too far" and "a hurdle too high" are idioms that mean "a task too difficult to achieve" where the adjectival phrase "too far"/ "too high" comes after the noun.]
In your example, "A mile deep" acts as an adjectival phrase: it describes the noun phrase "dirt road." That is partly why "deep" comes after "mile". In fact "mile" is not a noun in this particular situation: it is used as an adverb to give a "measure" of the adjective "deep" and describe "how deep" the dirt road is:
How deep is the dirt road?
It is a mile deep.
Note: here "a mile deep" means the road goes a distance of one mile into the forest. That is like 1.6 kilometres.