MWE:

\startuseMPgraphic{CountFrame}
  picture p; numeric w, h, o;
  p := textext.rt("\wrappedcurrentconstruction");
  w := OverlayWidth; h := OverlayHeight; o := BodyFontSize;
    p := p shifted (2o,h-ypart center p);
    draw p;
    path b;
    b := boundingbox p enlarged (o/10);
    drawoptions (withpen pencircle scaled 1pt withcolor black);
    draw (2o,h)--(0,h)--(0,0)--(w,0)--(w,h)--(xpart urcorner b,h);
    draw b;
    setbounds currentpicture to OverlayBox;
\stopuseMPgraphic

\defineoverlay
  [CountFrame]
  [\useMPgraphic{CountFrame}]

\defineframedtext
  [CountFramedText]
  [frame=off,
   background=CountFrame,
   offset=\bodyfontsize,
   width=\textwidth,
%   style={\switchtobodyfont[11pt,tt]}]
   bodyfont=11pt,
   style={\tt},
   ]

\def\FrameTitle#1%
  {\setMPtext{CountFrame}
  {\hbox spread 1em{\hss\strut#1\hss}}}\setMPtext{CountFrame}
  {}

\defineenumeration
  [Counter]
  [title=yes,
   alternative=empty,
   before=\startCountFramedText,
   after=\stopCountFramedText]

\starttext

\startchapter

  [title=first]

\startCounter
  [title={The First Title},
   reference={tit1}]
   Some verbatim text
\stopCounter

\startCounter
  [title={The Second Title},
   reference={tit2}]
   Some verbatim text
\stopCounter

Refer to \in{code fragment}[tit1] bla

\stopchapter

\stoptext