Answering on the question from the title:

How to do controlled rollout using Kubernetes deployment

You can create a Deployment that will have specific fields in its manifest that will configure the way Kubernetes handles it.

With the fields like: podAntiAffinity, requiredDuringSchedulingIgnoredDuringExecution you can ensure that Kubernetes will distribute the Pods equally across the cluster Nodes. You can read more about it by following below documentation:

• Kubernetes.io: Docs: Concepts: Scheduling eviction: Assign Pod to Node

Having in mind following rollout schedule:

You could use CI/CD tools (like for example Jenkins) to rollout (change) the amount of replicas of your Deployment across a specific schedule.

You could create a Jenkins pipeline with a deploy stage where you could put your own command with it's scheduler (or delay).

The example of such Deployment that could be used with Jenkins is following:

cat << EOF | kubectl apply -f -
apiVersion: apps/v1
kind: Deployment
metadata:
  name: nginx
spec:
  selector:
    matchLabels:
      app: nginx
  replicas: ${REPLICAS_COUNT}
  template:
    metadata:
      labels:
        app: nginx
    spec:
      affinity:
        podAntiAffinity:
          requiredDuringSchedulingIgnoredDuringExecution:
          - labelSelector:
              matchExpressions:
              - key: app
                operator: In
                values:
                - nginx
            topologyKey: "kubernetes.io/hostname"
      containers:
      - name: nginx
        image: nginx
EOF

This Deployment will assign Pods to the Nodes that aren't already having an already running replica of this Deployment (i.e. 1 Pod = 1 Node). If the amount of Pods exceeds the amount of Nodes they will remain in Pending state.

Additional resources:

• Jenkins.io: Doc: Pipeline: Tour: Environment
• Kubernetes.io: Docs: Concepts: Workloads: Controllers: Deployment