diff --git a/contrib/teamcity/build-configurations.py b/contrib/teamcity/build-configurations.py
--- a/contrib/teamcity/build-configurations.py
+++ b/contrib/teamcity/build-configurations.py
@@ -34,6 +34,7 @@
         self.build_directory = None
         self.junit_reports_dir = None
         self.test_logs_dir = None
+        self.jobs = (os.cpu_count() or 0) + 1
 
         self.project_root = PurePath(
             subprocess.run(
@@ -115,7 +116,7 @@
         self.environment_variables = {
             "BUILD_DIR": str(self.build_directory),
             "CMAKE_PLATFORMS_DIR": self.project_root.joinpath("cmake", "platforms"),
-            "THREADS": str(os.cpu_count() or 1),
+            "THREADS": str(self.jobs),
             "TOPLEVEL": str(self.project_root),
         }
 
@@ -178,10 +179,15 @@
                 "-DCMAKE_CXX_FLAGS=-Werror",
             ])
 
-        # Some generator flags
-        generator_flags = []
-        if self.config.get("fail_fast", False):
-            generator_flags.append("-k0")
+        # Get the generator, default to ninja
+        generator = self.config.get("generator", {})
+        generator_name = generator.get("name", "Ninja")
+        generator_command = generator.get("command", "ninja")
+        generator_flags = generator.get("flags", ["-k0"])
+
+        # Max out the jobs by default when the generator uses make
+        if generator_command == "make":
+            generator_flags.append("-j{}".format(self.jobs))
 
         # Handle cross build configuration
         cross_build = self.config.get("cross_build", None)
@@ -222,16 +228,14 @@
         self.build_steps.append(
             {
                 "bin": "cmake",
-                # TODO: let the generator be configurable
-                "args": ["-G", "Ninja", str(self.project_root)] + self.cmake_flags,
+                "args": ["-G", generator_name, str(self.project_root)] + self.cmake_flags,
             }
         )
 
         for target_group in targets:
             self.build_steps.append(
                 {
-                    # TODO: let the generator be configurable
-                    "bin": "ninja",
+                    "bin": generator_command,
                     "args": generator_flags + target_group,
                 }
             )
diff --git a/contrib/teamcity/build-configurations.yml b/contrib/teamcity/build-configurations.yml
--- a/contrib/teamcity/build-configurations.yml
+++ b/contrib/teamcity/build-configurations.yml
@@ -214,9 +214,13 @@
       QEMU_LD_PREFIX: /usr/arm-linux-gnueabihf
 
   build-make-generator:
-    script: builds/build-make-generator.sh
+    generator:
+      name: 'Unix Makefiles'
+      command: make
+      flags:
+        - '-k'
     templates:
-      - common_unix_artifacts
+      - check+secp256k1
     timeout: 1200
 
   build-master:
diff --git a/contrib/teamcity/builds/build-make-generator.sh b/contrib/teamcity/builds/build-make-generator.sh
deleted file mode 100755
--- a/contrib/teamcity/builds/build-make-generator.sh
+++ /dev/null
@@ -1,14 +0,0 @@
-#!/usr/bin/env bash
-
-export LC_ALL=C.UTF-8
-
-set -euxo pipefail
-
-# shellcheck source=../ci-fixture.sh
-source "${TOPLEVEL}/contrib/teamcity/ci-fixture.sh"
-
-# Ensure that the build using cmake and the "Unix Makefiles" generator is not
-# broken.
-git clean -xffd
-cmake -G "Unix Makefiles" "${TOPLEVEL}"
-make -j "${THREADS}" all check