To solve this problem, we first need to translate the natural language description into a symbolic representation. Let's denote the number of burgers as $x_1$ and the number of sausages as $x_2$. 

The objective function is to maximize profit, which can be represented algebraically as $5x_1 + 3x_2$, since each burger is sold for a profit of $5 and each sausage is sold for a profit of $3.

The constraints are:
- The total amount of ground meat used does not exceed 1000 grams: $20x_1 + 10x_2 \leq 1000$.
- At least three times the number of sausages are needed than burgers: $x_2 \geq 3x_1$.
- There needs to be at least 10 burgers made: $x_1 \geq 10$.
- The number of burgers and sausages cannot be negative: $x_1, x_2 \geq 0$.

Thus, the symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'number of burgers'), ('x2', 'number of sausages')],
    'objective_function': '5*x1 + 3*x2',
    'constraints': ['20*x1 + 10*x2 <= 1000', 'x2 >= 3*x1', 'x1 >= 10', 'x1 >= 0', 'x2 >= 0']
}
```

Now, let's write the Gurobi code to solve this linear programming problem:
```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(name='x1', vtype=GRB.CONTINUOUS, lb=0)
x2 = m.addVar(name='x2', vtype=GRB.CONTINUOUS, lb=0)

# Set the objective function
m.setObjective(5*x1 + 3*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(20*x1 + 10*x2 <= 1000, name='meat_limit')
m.addConstr(x2 >= 3*x1, name='sausage_to_burger_ratio')
m.addConstr(x1 >= 10, name='minimum_burgers')

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Number of burgers: {x1.x}")
    print(f"Number of sausages: {x2.x}")
    print(f"Maximum profit: ${5*x1.x + 3*x2.x:.2f}")
else:
    print("No optimal solution found.")
```