Nice job Fate!
Selective Color/White method: A quick way to color a blown-out sky (that also fills in the gaps in trees and branches, etc. - no picky selections necessary). No clouds, just solid blue sky.
1. LOOSELY select sky, making sure not to include other white/light non-sky areas (be sure to include trees & branches w/white sky in them)
2. go to Image>Adjustments>Selective Color (or use Selective Color adjustment layer found in the Layers menu)
Check Absolute at bottom
Choose Whites from the dropdown
3. Adjust Cyan, Magenta & Yellow sliders until get the blue shade you wish. (in your pic, choose a hue that matches the "blue" hills in background)
slide Cyan & Magenta to plus side (more Cyan than Magenta)
slide Yellow to 0 or to minus side.
slide Black to adjust brightness - or, I prefer to leave it a little over-saturated and use the Layers "opacity" to adjust the final brightness.
4. Use History Brush to undo any selection mistakes - often not needed at all.
5. adjust "opacity" in Layers window, if necessary.
6. This also works on skies that are half washed-out just at one side (half blue, half white) - you can even out the color to be a solid blue, or adjust to a half blue, half light blue - whichever you prefer.