There are 51 solutions to the alphametic using primes:

pie  end  dry              total
127  769  953               1849
127  769  983               1879
157  769  983               1909
163  359  947               1469
163  389  947               1499
257  719  983               1959
257  739  941               1937
257  769  941               1967
257  769  983               2009
263  347  719               1329
263  347  751               1361
263  349  971               1583
263  359  941               1563
263  359  947               1569
263  359  971               1593
263  379  941               1583
263  389  941               1593
263  389  947               1599
263  389  971               1623
263  397  751               1411
283  317  769               1369
283  347  719               1349
283  347  751               1381
283  347  761               1391
283  347  769               1399
283  349  967               1599
283  349  971               1603
283  359  941               1583
283  359  947               1589
283  359  967               1609
283  359  971               1613
283  367  719               1369
283  367  751               1401
283  379  941               1603
283  397  751               1431
283  397  761               1441
293  347  751               1391
293  347  761               1401
293  367  751               1411
457  719  983               2159
457  769  983               2209
467  719  953               2139
467  719  983               2169
487  719  953               2159
487  769  953               2209
547  719  983               2249
547  769  983               2299
587  739  941               2267
587  769  941               2297
647  719  953               2319
647  719  983               2349

The totals have not been tested for prime vs composite, but none seem to fit the criterion that all pairs of its digits form a 2-digit prime. Any total that has an even number (including zero) fails. Any that has two matching digits other than 1 fails. Any that has both a 3 and a 9 fails, or a 1 and a 5 for that matter.

DECLARE FUNCTION nodup! (x$)
DATA         101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179
DATA         181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269
DATA         271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367
DATA         373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461
DATA         463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571
DATA         577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661
DATA         673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773
DATA         787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883
DATA         887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997 

DATA     11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , 53 , 59 , 61 , 67 , 71
DATA     73 , 79 , 83 , 89 , 97

DIM prime(143), prime2(21)

FOR i = 1 TO 143: READ prime(i): PRINT prime(i); : NEXT

PRINT

FOR i = 1 TO 21: READ prime2(i): PRINT prime2(i); : NEXT

PRINT : PRINT

FOR p1 = 1 TO 141
 pie = prime(p1): pie$ = LTRIM$(STR$(pie))
 chk1$ = LEFT$(pie$, 2)
 IF nodup(pie$) THEN
FOR p2 = p1 + 1 TO 142
 end0 = prime(p2): end0$ = LTRIM$(STR$(end0))
 IF RIGHT$(pie$, 1) = LEFT$(end0$, 1) AND nodup(chk1$ + end0$) THEN
   chk2$ = chk1$ + LEFT$(end0$, 2)
FOR p3 = p2 + 1 TO 143
   dry = prime(p3): dry$ = LTRIM$(STR$(dry))
 IF RIGHT$(end0$, 1) = LEFT$(dry$, 1) AND nodup(chk2$ + dry$) THEN
   chk3$ = chk2$ + dry$
   IF nodup(chk3$ + "0") THEN
     tot = pie + end0 + dry
     IF tot > 999 AND tot < 10000 THEN
      t$ = LTRIM$(STR$(tot))
      good = 1
      FOR i = 1 TO 4
        FOR j = 1 TO 4
          IF i <> j THEN
            ptest = VAL(MID$(t$, i, 1) + MID$(t$, j, 1))
            foundp = 1
            FOR k = 1 TO 21
             ' IF ptest = prime2(k) THEN foundp = 1: EXIT FOR
            NEXT
            IF foundp = 0 THEN good = 0: EXIT FOR
          END IF
        NEXT j
        IF good = 0 THEN EXIT FOR
      NEXT i
      IF good THEN
          PRINT pie; end0; dry, tot
          ct = ct + 1: IF ct MOD 40 = 0 THEN DO: LOOP UNTIL INKEY$ > "": PRINT
      END IF
     END IF
   END IF
 END IF
NEXT p3
 END IF
NEXT p2
 END IF
NEXT p1
PRINT ct

FUNCTION nodup (x$)
  nd = -1
  FOR i = 1 TO LEN(x$) - 1
    IF INSTR(MID$(x$, i + 1), MID$(x$, i, 1)) THEN nd = 0: EXIT FOR
  NEXT
  nodup = nd
END FUNCTION

Note the commenting out of the tests for primality of the pairs so that the candidates could be listed.