Make me a loaf of bread before I'm fired!

Charcoal, 35 33 30 bytes

Saved 2 bytes thanks to @ASCII-only

Saved 3 bytes thanks to @Neil by rearranging the order in which the lines where drawn

ＮωＮηＮδ↓η↗→×_ω↑η←\←ω↗δ→/ω↓\↓η↙δ

Try it online!

This is my first try at Charcoal. I heard it was good at ascii-art, so I tried it out.

Explanation (outdated)

ＮωＮηＮδ                  # Take the width, height and depth as input
↗¹                        # Write 1 /
↗δ                        # Write depth times /
¶ω                        # Move the cursor down and write - width times
\                         # Write \
↙↓η                       # Move the cursor one down and one left, and then write | height times
↙δ                        # Write / depth times

Now the back part of the loaf is complete. Now comes the front face.

↑↑η                       # Go up and write | height times
←\                        # Write a \ and move 1 to the left
←ω                        # Write - width times
↓↓η                       # Go down and write | height times
↗Ｐ×_ω                    # Go northeast and write _ width times

C, 270 290 320 328 bytes

#define P printf(
p(n,s)char*s;{P s,--n&&p(n,s));}i,j,k;f(w,h,d){P "%*c",d+1,47);p(w,"-");P "\\\n");for(i=d,j=h,k=0;--i;)P "%*c%*c\n",i+1,47,w+d+1-i-k+!!k,j-->0?124:(++k>0)*47);P "/");p(w,"-");P "\\%*c\n",d-k,j-->0?124:47);for(k&&k++;++i<h;)P "|%*c%*c\n",w+1,124,d-k>!k?d+1-k-!k:0,j-->0?124:(++k>0)*47);P "|");p(w,"_");P "|/");}

Finally (after two three incorrect versions) it works correctly with large sizes also.

Try it online!

JS (ES6), 375 bytes

s=(s,t)=>s.repeat(t);(d,c,e)=>{z=" ",y=`
`,x="\\",w="/",v="|",u="-";a=s(z,e--);a=a+w+s(u,d);a+=x+y;for(b=0;b<e;b++)a+=s(z,e-b),b<c?(a+=w,a+=s(z,d+b+1),a+=v):(a+=w,a+=s(z,d+c+1),a+=w),a+=y;a=a+w+s(u,d);a+=x;f=0;b<c?(a+=s(z,e),a+=v):(a+=s(z,c-f),a+=w);b++;f++;a+=y;for(g=0;g<c;g++)a+=v,a+=s((g==c-1?"_":z),d),a+=v,b<c?(a+=s(z,e),a+=v):(a+=s(z,c-f),a+=w),b++,f++,a+=y;return a}

Ungolfed:

var proc = function(width, height, depth){
    var str = "";

    //Back of loaf (depth)
    str += " ".repeat(depth);//initial padding before start of back of loaf
    str += "/";//top left corner of back of loaf
    str += "-".repeat(width);//top edge of back of loaf
    str += "\\";//top left corner of back of loaf
    str += "\n";//end line

    //Depth lines
    for(var i = 0; i < depth - 1; i++){
        console.log(i)
        str += " ".repeat(depth - i - 1);//space padding before start of left middle edge
        if(i < height){
            //str += " ".repeat(depth - i - 2);
            str += "/";//left middle edge
            str += " ".repeat(width + i + 1);//middle of loaf
            str += "|";//end that edge
            str += "\n";//end line
        }else{
            str += "/";//left middle edge
            str += " ".repeat(width + height + 1);//between two /s
            str += "/";//right middle edge
            str += "\n";//end line
        }
    }

    //front top edge of loaf
    str += "/";//top left corner
    str += "-".repeat(width);//top edge
    str += "\\";//top right corner

    var i3 = 0;

    if(i < height){
        str += " ".repeat(depth - 1);//space for the incoming far right edge
        str += "|";//far right edge
        i++;
        i3++;
    }else{
        str += " ".repeat(height - i3);//space for the incoming far right edge
        str += "/";//far right edge
        i++;
        i3++;
    }

    str += "\n";//newline


    for(var i2 = 0; i2 < height; i2++){
        str += "|";//left edge
        str += (i2 === height - 1 ? "_" : " ").repeat(width);//if we are at the bottom, use underscores to mark that, otherwise spaces.
        str += "|";
        if(i < height){
            str += " ".repeat(depth - 1);//space for the incoming far right edge
            str += "|";//far right edge
            i++;
            i3++;
        }else{
            str += " ".repeat(height - i3);//space for the incoming far right edge
            str += "/";//far right edge
            i++;
            i3++;
        }
        str += "\n";//newline
    }

    return str;
};

s=(s,t)=>s.repeat(t);
var bakeBread = (d,c,e)=>{z=" ",y=`
`,x="\\",w="/",v="|",u="-";a=s(z,e--);a=a+w+s(u,d);a+=x+y;for(b=0;b<e;b++)a+=s(z,e-b),b<c?(a+=w,a+=s(z,d+b+1),a+=v):(a+=w,a+=s(z,d+c+1),a+=w),a+=y;a=a+w+s(u,d);a+=x;f=0;b<c?(a+=s(z,e),a+=v):(a+=s(z,c-f),a+=w);b++;f++;a+=y;for(g=0;g<c;g++)a+=v,a+=s((g==c-1?"_":z),d),a+=v,b<c?(a+=s(z,e),a+=v):(a+=s(z,c-f),a+=w),b++,f++,a+=y;return a}

<input type = "number" id = "a1">
<br/>
<input type = "number" id = "a2">
<br/>
<input type = "number" id = "a3">
<br/>
<button onclick = "document.getElementById('textarea').innerText = bakeBread(+(document.getElementById('a1').value), +(document.getElementById('a2').value), +(document.getElementById('a3').value))">Bake Bread!</button>
<br/>
<textarea id = "textarea"></textarea>