To solve this optimization problem, we first need to translate the given natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the provided information.

### Symbolic Representation

Let's define:
- `x1` as the number of cheeseburgers,
- `x2` as the number of hot dogs.

The objective function is to minimize: `3*x1 + 6*x2`.

Constraints are as follows:
1. Fat from cheeseburgers and hot dogs must be at least 11 grams: `5*x1 + 4*x2 >= 11`.
2. Calcium from cheeseburgers and hot dogs must be at least 8 milligrams: `2*x1 + 3*x2 >= 8`.
3. The constraint `10*x1 - 3*x2 >= 0` must be satisfied.
4. Total fat from cheeseburgers and hot dogs must not exceed 40 grams: `5*x1 + 4*x2 <= 40`.
5. Total calcium from cheeseburgers and hot dogs must not exceed 13 milligrams: `2*x1 + 3*x2 <= 13`.

### Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'cheeseburgers'), ('x2', 'hot dogs')],
    'objective_function': '3*x1 + 6*x2',
    'constraints': [
        '5*x1 + 4*x2 >= 11',
        '2*x1 + 3*x2 >= 8',
        '10*x1 - 3*x2 >= 0',
        '5*x1 + 4*x2 <= 40',
        '2*x1 + 3*x2 <= 13'
    ]
}
```

### Gurobi Code

Now, let's implement this optimization problem using Gurobi in Python. Note that we need to have Gurobi installed (`pip install gurobipy`) and a valid license to use it.

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define variables
x1 = m.addVar(lb=0, ub=GRB.INFINITY, vtype=GRB.CONTINUOUS, name="cheeseburgers")
x2 = m.addVar(lb=0, ub=GRB.INFINITY, vtype=GRB.INTEGER, name="hot_dogs")

# Define the objective function
m.setObjective(3*x1 + 6*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(5*x1 + 4*x2 >= 11, "fat_min")
m.addConstr(2*x1 + 3*x2 >= 8, "calcium_min")
m.addConstr(10*x1 - 3*x2 >= 0, "custom_constraint")
m.addConstr(5*x1 + 4*x2 <= 40, "fat_max")
m.addConstr(2*x1 + 3*x2 <= 13, "calcium_max")

# Optimize the model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Cheeseburgers: {x1.x}")
    print(f"Hot dogs: {x2.x}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found")
```