You don't need a calendar table for minutes because there's nothing really special about any minute between two points in time (unless you need to compensate for the night of a change to or from DST, but let's pretend everyone stores UTC because... that makes much more sense).

We can generate a dynamic numbers table using recursion (or sub in a numbers table you already have, or generate one using a variety of other methods that are easy to search for). We only need the largest gap, so we can determine that up front. Then we just join to all of the "minutes to add" that are less than or equal to the minute gap between any row's From / To values.

DECLARE @maxDelta int;

SELECT @maxDelta = MAX(DATEDIFF(MINUTE, UnavailableFrom, UnavailableTo))
FROM dbo.UnavailableTime;

WITH m(m) AS
(
  SELECT 0 UNION ALL SELECT m+1
  FROM m WHERE m < @maxDelta
)
SELECT u.UnavailableID, 
  MinuteBreakdown = DATEADD(MINUTE, m.m, u.UnavailableFrom)
FROM dbo.UnavailableTime AS u
INNER JOIN m ON m.m <= DATEDIFF(MINUTE, u.UnavailableFrom, u.UnavailableTo)
ORDER BY UnavailableID, MinuteBreakdown
OPTION (MAXRECURSION 32767);

• Example db<>fiddle

This works until you get to about 22 days of outage, then you need to change 32767 to 0.

This can do the trick (as long as you don't have many, many rows, First I calculate the number of minutes between UnavailableFrom and UnavailableTo. then for each UnavailableID I create a numerator till the number of minutes

SELECT UnavailableID, MinuteBreakdown 
FROM MyTable
CROSS APPLY (SELECT DATEDIFF(MINUTE, UnavailableFrom,UnavailableTo) MinuteCount ) ca
CROSS APPLY (SELECT TOP (MinuteCount) DATEADD(MINUTE, ROW_NUMBER() OVER( ORDER BY (SELECT NULL))-1, UnavailableFrom ) MinuteBreakdown FROM sys.all_objects) ca2

You can use Common Table Expressions :

;WITH CTE AS
(
  -- initial part
  SELECT UnavailableID, UnavailableFrom AS MinuteBreakdown, UnavailableFrom, UnavailableTo
  FROM TimeMachine
  WHERE UnavailableFrom < UnavailableTo
  UNION ALL
  -- recursive part
  SELECT UnavailableID, DATEADD(MI, 1, MinuteBreakdown), UnavailableFrom, UnavailableTo
  FROM CTE
  WHERE DATEADD(MI, 1, MinuteBreakdown) <= UnavailableTo
)
SELECT *
FROM CTE
ORDER BY UnavailableID, MinuteBreakdown;

SQL Fiddle