To solve this optimization problem, we need to define the decision variables, the objective function, and the constraints.

Let's denote:
- $x$ as the number of small chessboards produced,
- $y$ as the number of large chessboards produced.

The profit per small chessboard is $4, and per large chessboard is $8. Therefore, the total profit (which we want to maximize) can be represented by the objective function:
\[ \text{Maximize} \quad 4x + 8y \]

The constraints are based on the available time for cutting and assembly:
- Each small chessboard takes 5 minutes of cutting, and each large chessboard takes 10 minutes. The total available time for cutting is 400 minutes. So, we have:
\[ 5x + 10y \leq 400 \]
- Each small chessboard takes 10 minutes of assembly, and each large chessboard takes 20 minutes. The total available time for assembly is 700 minutes. Thus, we have:
\[ 10x + 20y \leq 700 \]

Additionally, $x$ and $y$ must be non-negative since they represent the number of chessboards.

Now, let's translate this problem into Gurobi code in Python:

```python
from gurobipy import *

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

# Define the decision variables
x = m.addVar(vtype=GRB.INTEGER, name="small_chessboards")
y = m.addVar(vtype=GRB.INTEGER, name="large_chessboards")

# Set the objective function to maximize profit
m.setObjective(4*x + 8*y, GRB.MAXIMIZE)

# Add constraints for cutting and assembly time
m.addConstr(5*x + 10*y <= 400, "cutting_time")
m.addConstr(10*x + 20*y <= 700, "assembly_time")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution: Produce {x.x} small chessboards and {y.x} large chessboards.")
else:
    print("No optimal solution found")
```