diff --git a/doc/source/whatsnew/v2.1.2.rst b/doc/source/whatsnew/v2.1.2.rst
index 8863bfa9f3f69..1dd7bc49fad42 100644
--- a/doc/source/whatsnew/v2.1.2.rst
+++ b/doc/source/whatsnew/v2.1.2.rst
@@ -26,6 +26,7 @@ Fixed regressions
 Bug fixes
 ~~~~~~~~~
 - Fixed bug in :class:`.DataFrameGroupBy` reductions not preserving object dtype when ``infer_string`` is set (:issue:`55620`)
+- Fixed bug in :meth:`.DataFrameGroupBy.min()` and :meth:`.DataFrameGroupBy.max()` not preserving extension dtype for empty object (:issue:`55619`)
 - Fixed bug in :meth:`.SeriesGroupBy.value_counts` returning incorrect dtype for string columns (:issue:`55627`)
 - Fixed bug in :meth:`Categorical.equals` if other has arrow backed string dtype (:issue:`55364`)
 - Fixed bug in :meth:`DataFrame.__setitem__` not inferring string dtype for zero-dimensional array with ``infer_string=True`` (:issue:`55366`)
diff --git a/pandas/core/arrays/base.py b/pandas/core/arrays/base.py
index 05e6fc09a5ef6..3d97711d5f8c3 100644
--- a/pandas/core/arrays/base.py
+++ b/pandas/core/arrays/base.py
@@ -2352,6 +2352,9 @@ def _groupby_op(
         # GH#43682
         if isinstance(self.dtype, StringDtype):
             # StringArray
+            if op.how not in ["any", "all"]:
+                # Fail early to avoid conversion to object
+                op._get_cython_function(op.kind, op.how, np.dtype(object), False)
             npvalues = self.to_numpy(object, na_value=np.nan)
         else:
             raise NotImplementedError(
diff --git a/pandas/core/groupby/ops.py b/pandas/core/groupby/ops.py
index 607059e5183ec..e4cba7ce8f1cd 100644
--- a/pandas/core/groupby/ops.py
+++ b/pandas/core/groupby/ops.py
@@ -33,6 +33,7 @@
 from pandas.errors import AbstractMethodError
 from pandas.util._decorators import cache_readonly
 
+from pandas.core.dtypes.base import ExtensionDtype
 from pandas.core.dtypes.cast import (
     maybe_cast_pointwise_result,
     maybe_downcast_to_dtype,
@@ -837,10 +838,8 @@ def agg_series(
         -------
         np.ndarray or ExtensionArray
         """
-        # test_groupby_empty_with_category gets here with self.ngroups == 0
-        #  and len(obj) > 0
 
-        if len(obj) > 0 and not isinstance(obj._values, np.ndarray):
+        if not isinstance(obj._values, np.ndarray):
             # we can preserve a little bit more aggressively with EA dtype
             #  because maybe_cast_pointwise_result will do a try/except
             #  with _from_sequence.  NB we are assuming here that _from_sequence
@@ -849,11 +848,18 @@ def agg_series(
 
         result = self._aggregate_series_pure_python(obj, func)
 
-        npvalues = lib.maybe_convert_objects(result, try_float=False)
-        if preserve_dtype:
-            out = maybe_cast_pointwise_result(npvalues, obj.dtype, numeric_only=True)
+        if len(obj) == 0 and len(result) == 0 and isinstance(obj.dtype, ExtensionDtype):
+            cls = obj.dtype.construct_array_type()
+            out = cls._from_sequence(result)
+
         else:
-            out = npvalues
+            npvalues = lib.maybe_convert_objects(result, try_float=False)
+            if preserve_dtype:
+                out = maybe_cast_pointwise_result(
+                    npvalues, obj.dtype, numeric_only=True
+                )
+            else:
+                out = npvalues
         return out
 
     @final
diff --git a/pandas/tests/groupby/test_reductions.py b/pandas/tests/groupby/test_reductions.py
index fdfb211ac2269..35ad8e3f5dc61 100644
--- a/pandas/tests/groupby/test_reductions.py
+++ b/pandas/tests/groupby/test_reductions.py
@@ -575,6 +575,19 @@ def test_groupby_min_max_categorical(func):
     tm.assert_frame_equal(result, expected)
 
 
+@pytest.mark.parametrize("func", ["min", "max"])
+def test_min_empty_string_dtype(func):
+    # GH#55619
+    pytest.importorskip("pyarrow")
+    dtype = "string[pyarrow_numpy]"
+    df = DataFrame({"a": ["a"], "b": "a", "c": "a"}, dtype=dtype).iloc[:0]
+    result = getattr(df.groupby("a"), func)()
+    expected = DataFrame(
+        columns=["b", "c"], dtype=dtype, index=pd.Index([], dtype=dtype, name="a")
+    )
+    tm.assert_frame_equal(result, expected)
+
+
 def test_max_nan_bug():
     raw = """,Date,app,File
 -04-23,2013-04-23 00:00:00,,log080001.log