First, let’s get accessories out of the way.
I’ve been unable to find a tutorial for these specific goggles, but your favourite steampunk goggle tutorial should work for this. Look especially at the ones that are altered swim goggles as the shape is very similar. The musical shows the rims to be silver with black leather around the eyepieces and for the strap. There are a couple of buckles for adjustments on either side near the front.
In addition to the goggles, I suggest carrying a cigarette. HERE is a good tutorial I found on youtube, but there are many other great tutorials out there as well!
Other suggested accessories would be a flamethrower or some cooking utensils – just don’t go with plastic or teflon as it’s not period. They’d look good either carried or shoved into the tie. Dynamite or a machine gun would also be fun.
The chef’s jacket is white, with the sleeves rolled up nearly to the elbow. I believe it’s only waist length. The apron goes over the bottom edge. The jacket is double-breasted with six to eight fabric-covered white buttons.
The collar has a notch on either side. He’s got a tendency to leave it flipped up in the anime although the musical’s collar is shaped in such a way to be permanently stand-up. There is matching white piping around the edges and across the front on the far side of the buttons.
It could be altered from a purchased jacket.
The apron falls past the knees and goes all the way around with wide straps long enough to wrap around the waist to tie in front with only short ends left. He ties them in a small, even bow. The width in the musical has been exaggerated from the anime to be more visible on stage and also to help the apron stay on. Your choice on width.
The apron is probably made of a rectangle with a waistband at the top (which the tie is attached to) but I’m not entirely sure of the precise shape. It may be angled away. Note how the top overlaps in the back above.
Under the apron he wears matching plain white pants. They tuck into his boots.
His boots are good solid black work boots with grey treads. (Possibly Wellingtons) They’re probably rubber. Note that they look like soldier boots, which fits with Bard’s origin.