To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. This involves defining variables for the quantities of small and large PCBs to be produced, formulating the objective function based on the profits per unit, and translating the constraints related to silicon availability, design time, soldering time, and minimum production requirements.

Let's denote:
- \(x_1\) as the number of small PCBs,
- \(x_2\) as the number of large PCBs.

The objective function aims to maximize profit. Given that the profit per small PCB is $20 and per large PCB is $35, we can write it as:
\[ \text{Maximize: } 20x_1 + 35x_2 \]

The constraints are as follows:
1. Silicon availability constraint: \(3x_1 + 5x_2 \leq 250\)
2. Design time constraint: \(30x_1 + 40x_2 \leq 800\)
3. Soldering time constraint: \(20x_1 + 30x_2 \leq 600\)
4. Minimum production constraints:
   - For small PCBs: \(x_1 \geq 5\)
   - For large PCBs: \(x_2 \geq 6\)

All variables are non-negative since they represent quantities of items.

Thus, the symbolic representation of the problem is:
```json
{
  'sym_variables': [('x1', 'number of small PCBs'), ('x2', 'number of large PCBs')],
  'objective_function': '20*x1 + 35*x2',
  'constraints': [
    '3*x1 + 5*x2 <= 250',
    '30*x1 + 40*x2 <= 800',
    '20*x1 + 30*x2 <= 600',
    'x1 >= 5',
    'x2 >= 6'
  ]
}
```

Now, let's write the Gurobi code in Python to solve this linear programming problem:

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="small_PCBs")
x2 = m.addVar(vtype=GRB.INTEGER, name="large_PCBs")

# Set the objective function
m.setObjective(20*x1 + 35*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(3*x1 + 5*x2 <= 250, "silicon")
m.addConstr(30*x1 + 40*x2 <= 800, "design_time")
m.addConstr(20*x1 + 30*x2 <= 600, "soldering_time")
m.addConstr(x1 >= 5, "min_small_PCBs")
m.addConstr(x2 >= 6, "min_large_PCBs")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Small PCBs: {x1.x}")
    print(f"Large PCBs: {x2.x}")
    print(f"Maximum Profit: {m.objVal}")
else:
    print("No optimal solution found")
```