Disclaimer: I Do Not Own Doctor Who
He knows that he should go searching for someone who could heal him, but he isn't ready for that yet so he searched for the person who made all his suffering possible. The Doctor followed pointless chaos and destruction to find Missy because everyone knew if you followed those things long enough you would find her.
The Doctor wasn't sure why he wanted to see her because as tempting as it was to cause the same hurt she had caused him he wouldn't or couldn't do the same thing to her. Missy would love it if he did, it would prove to her that they were both the same. It's what she had craved for years for them to be the same and he has never known why.
He knows that going to find her will feed one of her future attempts to hurt him, but it doesn't stop him following her trail of destruction.
He walks through a city turned into nothing but rubble on a planet where the sand is blue and the sky violet. It takes him hardly any time at all to find Missy, she's twirling around and around taking great delight in what she has done.
The Doctor is tempted to open his mouth and start lecturing her about the value of life, but it wouldn't mean anything to her so he doesn't bother because it's a lost cause.
He isn't sure how long it takes for her to look at him and acknowledge he is there, time is meaningless to him now.
Missy smiles at him and he isn't sure whether because she's genuinely happy to see him or if she's just thinking of all the things that she will say to taunt him.
"Are you ready to admit that deep down we are both the same yet?" Missy asked him, her head tilted to one side slightly in curiosity for his answer.
"We are not the same Missy." He sighed, Missy pouts at his answer almost like a child whose has been told they have to eat their vegetables.
"One day you will learn that we are and then everything will be okay again," Missy answered, madness dancing in her eyes. "It's like when we were children and you wouldn't believe that I could run further than you until I beat you every time we raced."
"You cheated, that's why you always won." The Doctor said, scowling at the amount times he was tripped or shoved to the ground.
"I did not!" Missy whined, it was the same tone she used when they were children. "Just to prove it, I will race you to the top of that hill."
The Doctor didn't come here to run like they were children, he wasn't sure exactly what he'd come here for.
He can see the challenge in Missy's eyes and he knows that he should refuse that challenge but he doesn't.
Like when they were children they run both trying to win the race.