Egwene is noting Mesaana's strength of will in TAR when she makes the comment about not daring to face her alone ... she nearly lost her mind just minutes prior

As for Egwene facing Mesaana ... there are several comments around Egwene's plans and then there is the fact that she was alone when she thought Mesaana was coming for her.

Mesaana would need something grand to offset so many failures. Killing Egwene would work. That might send the White Tower back into division.

Gawyn had been mortified when she'd said she might use herself as bait. Dared she do so? She gripped the railing, standing above the Tower, above the city that depended on her, looking out on a world that needed her.

Something had to be done; Mesaana had to be drawn out. If what Saerin said was true, then the woman would be willing to fight directly—she wouldn't hide and poke from the shadows. Egwene's task, then, was to tempt her with an opportunity, one that didn't seem obvious, one she couldn't resist.

a plan which she shared with NO ONE

Then there is this:

She folded her arms beneath her breasts. He could almost feel the heat of her anger. "Your yelling has drawn half of the White Tower. They saw you captured. The assassin probably knows, now, about my weaves."

"Light, Egwene! You talk as if I did it on purpose. I was only trying to protect you."

"I didn't ask for your protection! I asked for your obedience! Gawyn, don't you see the opportunity we've missed? If you hadn't scared Mesaana away, she'd have walked into my traps!"

Making it even more clear that Egwene had a plan all her own ... you know as well as I do that an Amyrlin is not allowed to willingly put herself in danger so the Hall would never have approved of this.

then we have her direct statement:

"I have sisters researching everything we can find about Mesaana," Egwene continued. "She's not a warrior, Gawyn. She's an administrator, a planner. If I can confront her, I can defeat her. But we must find her first. Exposing myself is only one of my plans—and you are right, it is dangerous. But my precautions have been extensive."

Now consider that when Gawyn sets off her wards she is alone, no one else is in the room, just Egwene, waiting for Mesaana. She clearly thinks she can confront Mesaana 1:1 and come out victorious.

Personally I think she's being foolish, she's likely thinking in terms of their strength being close enough for Egwene to have a chance against a woman who is not a warrior, but realistically Mesaana should be far more skilled than Egwene.

In the end Mesaana was stupid and arrogant and played into Egwene's strength in TAR rather than attempt an OP duel ... odd, if as you say, she was so much stronger than Egwene, she could have easily finished the job this way. Why risk confronting a Dreamer in TAR when you should be able to manhandle her with your superior OP strength and knowledge?