To solve this optimization problem, we first need to define the decision variables and the objective function. Let's denote:
- \(x\) as the number of small puzzles made,
- \(y\) as the number of large puzzles made.

The objective is to maximize profit. Given that the profit per small puzzle is $8 and per large puzzle is $12, the total profit \(P\) can be represented by the equation:
\[ P = 8x + 12y \]

We have two main constraints based on the time available for cutting and quality checking:
1. Cutting time constraint: Each small puzzle takes 10 minutes of cutting, and each large puzzle takes 15 minutes. The company has 600 minutes available for cutting. This can be represented as:
\[ 10x + 15y \leq 600 \]
2. Quality checking time constraint: Each small puzzle requires 20 minutes of quality checking, and each large puzzle requires 30 minutes. With 1000 minutes available for quality checking, we have:
\[ 20x + 30y \leq 1000 \]

Additionally, \(x\) and \(y\) must be non-negative since the company cannot produce a negative number of puzzles.

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

```python
from gurobipy import *

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

# Define decision variables
x = m.addVar(lb=0, vtype=GRB.INTEGER, name="small_puzzles")
y = m.addVar(lb=0, vtype=GRB.INTEGER, name="large_puzzles")

# Set the objective function to maximize profit
m.setObjective(8*x + 12*y, GRB.MAXIMIZE)

# Add constraints
m.addConstr(10*x + 15*y <= 600, "cutting_time")
m.addConstr(20*x + 30*y <= 1000, "quality_checking_time")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution found: Produce {x.x} small puzzles and {y.x} large puzzles.")
else:
    print("No optimal solution found. The problem might be infeasible.")

```
```python
from gurobipy import *

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

# Define decision variables
x = m.addVar(lb=0, vtype=GRB.INTEGER, name="small_puzzles")
y = m.addVar(lb=0, vtype=GRB.INTEGER, name="large_puzzles")

# Set the objective function to maximize profit
m.setObjective(8*x + 12*y, GRB.MAXIMIZE)

# Add constraints
m.addConstr(10*x + 15*y <= 600, "cutting_time")
m.addConstr(20*x + 30*y <= 1000, "quality_checking_time")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution found: Produce {x.x} small puzzles and {y.x} large puzzles.")
else:
    print("No optimal solution found. The problem might be infeasible.")

```