Assuming there are no catch, I will go pure logic on this.
Step 1 : E = 0.
Explanation, E + E = E is only true for E = 0.
Result : PUZZL0 + PUZZL0 = Y0ARON0
Step 2 : Y = 1.
Explanation, No number doubled could give an extra 0 and end by a 2 or more.
Result : PUZZL0 + PUZZL0 = 10ARON0
Step 3 : P = 5.
Explanation, only 5 + 5 can give 10.
Result : 5UZZL0 + 5UZZL0 = 10ARON0
-U must be 2,3 or 4 or else the number would end in 11.
-Z must be under 5 or else both Z additions would add up to the same value. So it can be 2,3,4.
-A must therefore be 4,6 or 8
-N is even because there are no carry over from the addition of 0. So it can be 4,6 or 8.
-L must therefore be 2,3,4,7,8 or 9.
-N must therefore be over 4 to make the following Z addition +1. So can be 6,7,8,9.
-R and O can therefore only be (4,6,8)(7,9)
-Z can't be 2 or else one of the 2 Z additions would end in 5.
-Z can't be 4 or else A would be forced to be 6 and R would be 8, so no value left for N.
Step 4 : Z = 3 and therefore R = 6 and O = 7.
Explanation, only possible value left for Z.
Result : 5U33L0 + 5U33L0 = 10A67N0
Only 2,4,8,9 left to use.
Possible combinations for U+U = A are 2-4 and 4-8.
N must be 4 or 8. 4 is impossible because it would block both combinations.
Step 5 : U = 2, A = 4 and N = 8.
Explanation, only possible combination for (U,A,N).
Result : 523390 + 523390 = 1046780
E must be 0 or 5.
Y must be 1 or 2.
Z must be 0,1,2,3,7,8,9 or else both Z additions would add to the same value.
P must be 3,4,5,6,7,8 or 9 or else Y would be 0.