To generate test cases for the problem "DAG Trio," we need to create a random connected directed graph with the given constraints. Here is a Python function to generate test cases for this problem:

```python
import random
from collections import defaultdict

def generate_connected_graph(N, M):
    # Generate a connected graph using a variation of the Kruskal's algorithm
    
    # Initialize parent and rank for union-find
    parent = list(range(N))
    rank = [0] * N
    
    def find(x):
        if parent[x] != x:
            parent[x] = find(parent[x])
        return parent[x]
    
    def union(x, y):
        rootX = find(x)
        rootY = find(y)
        if rootX != rootY:
            if rank[rootX] > rank[rootY]:
                parent[rootY] = rootX
            elif rank[rootX] < rank[rootY]:
                parent[rootX] = rootY
            else:
                parent[rootY] = rootX
                rank[rootX] += 1
    
    edges = set()
    while len(edges) < M:
        # Randomly select two different vertices
        u, v = random.sample(range(1, N + 1), 2)
        if (u, v) not in edges and (v, u) not in edges:
            edges.add((u, v))
            union(u - 1, v - 1)
    
    return list(edges)

def tcgen():
    N = random.randint(3, 500)
    M = random.randint(max(3, N - 1), 30000)
    
    edges = generate_connected_graph(N, M)
    
    # Ensure that the graph is connected
    while not is_connected(N, edges):
        edges = generate_connected_graph(N, M)
    
    return N, M, edges

def is_connected(N, edges):
    # Check if the graph is connected using BFS/DFS
    adj_list = defaultdict(list)
    for u, v in edges:
        adj_list[u].append(v)
        adj_list[v].append(u)
    
    visited = set()
    def dfs(node):
        stack = [node]
        while stack:
            current = stack.pop()
            if current not in visited:
                visited.add(current)
                stack.extend(adj_list[current])
    
    # Start DFS from node 1
    dfs(1)
    
    return len(visited) == N

# Example of generating a test case
if __name__ == "__main__":
    N, M, edges = tcgen()
    print(N, M)
    for edge in edges:
        print(edge[0], edge[1])
```

### Explanation:

1. **Graph Generation**: The `generate_connected_graph` function generates a connected graph using a variation of Kruskal's algorithm. It ensures the graph is connected by using a union-find data structure to guarantee that all vertices are in the same connected component.

2. **Graph Validation**: The `is_connected` function checks if the graph is connected using DFS. It ensures that all vertices are reachable from any starting vertex.

3. **Main Function**: The `tcgen` function generates random values for \(N\) and \(M\) within the given constraints and ensures that the generated graph is connected by re-generating it until it meets the connectivity criteria.

4. **Test Case Output**: The generated test case includes the number of vertices \(N\), the number of edges \(M\), and the list of edges.

This generator ensures that the graph is connected and adheres to the problem constraints, producing valid and varied test cases for the "DAG Trio" problem.