Sure, let's create a test case generator for the given problem statement. We'll follow these steps:

1. Generate a random value for \(N\) within the specified constraints.
2. Generate a random value for \(M\) within the specified constraints.
3. Generate a sorted list of \(M\) unique integers \(A_i\) within the range \([2, 2^N]\).

Here is the test case generator code:

```python
import random

def tcgen():
    # Generate N within the range [1, 16]
    N = random.randint(1, 16)
    
    # Generate M within the range [0, 16]
    M = random.randint(0, 16)
    
    # Generate M unique integers A_i within the range [2, 2^N]
    if M > 0:
        A = sorted(random.sample(range(2, 2**N + 1), M))
    else:
        A = []
    
    return N, M, A

# Example usage
if __name__ == "__main__":
    for _ in range(5): # Generating 5 test cases as an example
        N, M, A = tcgen()
        print(f"{N} {M}")
        if M > 0:
            print(" ".join(map(str, A)))
        else:
            print("")
```

### Explanation:
1. **N Generation**: We generate \(N\) using `random.randint(1, 16)`.
2. **M Generation**: We generate \(M\) using `random.randint(0, 16)`.
3. **Generating A**: We use `random.sample(range(2, 2**N + 1), M)` to generate \(M\) unique integers from the range [2, 2^N]. If \(M\) is 0, we just assign an empty list to \(A\).
4. **Sorting A**: Since \(A\) needs to be in strictly increasing order, we use `sorted(...)`.

This generator will create valid test cases according to the problem constraints. You can modify the number of test cases generated in the example usage section as needed.