Is it possible to "curry" higher-kinded types in Scala?

You mean something like this?

trait QuxWithString[A] extends Qux[A, String]
new Turkle[QuxWithString]{}

This is the analog to partial application for types.

trait Turkle[C[_]]
trait Qux[A,B]
trait Wraps[A] {
  type Jkz[X] = Qux[A,X]
  trait Baz extends Turkle[Jkz]
}

Jason Zaugg came up with the most succinct way to do this:

trait Baz[A] extends Turkle[({type x[a]=Qux[A, a]})#x]

IntelliJ's Scala plugin will optionally collapse this to:

trait Baz[A] extends Turkle[x[a]=Qux[A, a]]

The compiler plugin kind projector allows this as well:

// Explicit lambda, greek letters
trait Baz[A] extends Turkle[λ[α=>Qux[A,α]]]

// Explicit lambda, normal letters
trait Baz[A] extends Turkle[Lambda[a=>Qux[A,a]]]

// No explicit lambda, ? placeholder    
trait Baz[A] extends Turkle[Qux[A,?]]