Sure! Below is a Python function to generate test cases for the problem "Fissure Puzzle Hard." The function generates a random grid that fits the problem's constraints.

```python
import random

def generate_fissure_puzzle_hard_test_case():
    # Generate an odd number N between 3 and 2047
    N = random.choice([i for i in range(3, 2048) if i % 2 != 0])
    
    # Generate the grid of NxN where each cell is either 'o' or 'x'
    grid = []
    for _ in range(N):
        row = ''.join(random.choice(['o', 'x']) for _ in range(N))
        grid.append(row)
    
    return N, grid

def format_test_case(N, grid):
    # Format the test case into the required input format as a string
    test_case_str = f"{N}\n"
    test_case_str += '\n'.join(grid)
    return test_case_str

# Example of generating a test case and printing it
N, grid = generate_fissure_puzzle_hard_test_case()
test_case_str = format_test_case(N, grid)
print(test_case_str)
```

### Explanation:
1. **Generating N:** The code first generates an odd number \( N \) between 3 and 2047 using `random.choice` to ensure the number is odd.
2. **Generating the Grid:** For each row in the \( N \times N \) grid, we generate a string of length \( N \) where each character is randomly chosen to be either 'o' (white) or 'x' (black).
3. **Formatting the Output:** The function `format_test_case` converts the generated grid into the required input format as a multiline string.

You can use this generator to produce random test cases for the problem. Each execution of the generator will provide a different test case within the specified constraints.