vnext typing update (#36509)

Author: kristapratico

eng/tox/tox.ini

@@ -170,7 +170,7 @@ commands =
 
 [testenv:next-mypy]
 description=Typechecks a package with the latest version of mypy
-mypy_version=1.9.0
+mypy_version=1.10.1
 skipsdist = true
 skip_install = true
 usedevelop = true
@@ -219,7 +219,7 @@ commands =
 
 [testenv:next-pyright]
 description=Typechecks a package with the latest version of static type-checker pyright
-pyright_version=1.1.359
+pyright_version=1.1.371
 skipsdist = true
 skip_install = true
 usedevelop = true